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369

Interferometric Measurements of

The Complex Dielectric

Constant of Liquids

%.

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U.S. DEPARTMENT OF COMMERCE

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NBS TECHNICAL NOTE 369

ISSUED AUGUST 1968

INTERFEROMETRIC MEASUREMENTS OF THE

COMPLEX DIELECTRIC CONSTANT OF LIQUIDS

W. S. LOVELL AND L. M. THIEL

Radio Standards Physics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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Price 50 cents.

TABLE OF CONTENTS

Page

Paper I - Errors in the "Perfect Square" Approximation 1 Abstract 1

1. Introduction 2

2. The Approximation 6 = tt 3

3. The Determination of e

' and e

" 5

4. Nonlinearities in -ln[ exp ( -na\ )] vs n 6

5. Errors in the Perfect Square Approximation 8

6. Conclusions 8

Appendix: Determination of Local Extrema 10

Acknowledgement 11

References 12

Figures 13

Paper II - Experiment Apparatus 19

Abstract 19

1. Introduction 20

2. Apparatus 20

3. Operation 24

Dielectric Thickness Determination 24

Power Level Determination 26

4. Instrumental Errors 29

Standing Wave Effects 30

Diffraction Effects 31

Beam Displacement Effects 33

Multiple Reflection Effects 35

Error Minimization 3 7

5. Summary .40

Acknowledgements .41

References .42

Figures .43

in
Table of Contents - Continued

Page

Paper III - Derivation of the Absolute Reflection Coefficient 51

Abstract 51

1. Introduction 52

2. Derivation for the Ideal Case 53

3. The Finite Sample and/or Receiving Surface 55

4. Derivation to Include a Distribution in X q and 6q .58

5. Summary 60

References . 62

Figures 63

Paper IV - A Numerical Method for Determining the Best Complex 65

Dielectric Constant

Abstract 65

1. Introduction 66

2. The Iteration Functions 68

Generalized Newton Iteration Function 69

Least Squares Iteration Function 71

3. Determination of the Vector Solution. 74

Analysis of the Integration 75

Numerical Integration for a Double Integral . 75

Differentiation Under the Integral Sign 76

Application of the Iteration Functions .76

Initial or Given Parameters 77

Second Approximation to the Vector Solution 79

Final Vector Solution 80

4. Accuracy of the Vector Solution 80

5. Discussion 81

Appendix: An Efficient Matrix Evaluation 85

References 86

IV
Interferometric Measurements of the

Complex Dielectric Constant of Liquids

I. Errors in the "Perfect Square" Approximation

William S. Lovell

ABSTRACT

Errors arising from the use of a "perfect square"

approximation in treating free- space, interferometric measurements of the complex dielectric constant e* = €' - ic " of

liquids are described. It is shown that such errors (1) are

generally positive, (2) often exceed values previously estimated

for the measurement technique as a whole, and (3) are not generally predictable without some prior knowledge of the dielectric

properties of the material being investigated. In addition,

these errors depend upon the dielectric thickness over which

data are incorporated into the calculations. Finally, an anomaly

in the calculations previously attributed to experimental error

is shown to arise from the "perfect square" approximation itself.

It is concluded that this approximation may at best provide initial

estimates for e' and e" which may then be used in a more complete

mathematical treatment.
1. INTRODUCTION

The reflection of electromagnetic radiation at oblique incidence in

free- space from an air-dielectric-metal configuration has been used to

determine the complex dielectric constant c* - €* - ic" of liquids at milli1-4

meter wavelengths. The reflected power R as a function of the dielectric

sheet thickness d yields a decaying interference pattern that has been

described by the equation

2 o n fl j , « -2 ad -4a d

_, p - 2p cos (2ffd + 6)e + e

is. =

' . (1

)

-> /->/. j ex -2ad 2 -4ad

1 - 2qcos (28 d - 6) e +p e

This expression represents the modulus squared of the complex reflection

coefficient for the configuration described. Reflection at the air-dielectric

interface is described by the complex Fresnel reflection coefficient pe

for the appropriate polarization, where P and 6 are functions of e*, the angle

of incidence 9 , and the free- space wavelength X of the incident radiation.

1 -4

The applications described have been based on use of the coefficient for

radiation whose electric vector lies perpendicular to the plane of incidence.

At the dielectric-metal interface, reflection has been assumed to be perfect

(reflection coefficient = -1) for a silver-plated metal. Propagation within

the dielectric in a direction normal to the parallel air-dielectric and

dielectric-metal interfaces is described by a complex propagation constant

k cos 6 = a + i#, where Q is the angle of refraction.

Application of Eq. (1) to the determination of complex dielectric

constants requires (1) demonstration that the equation describes experimental

data with sufficient accuracy and (2) adequate mathematical treatment of the

equation itself. Such a demonstration has not been given since complex

dielectric constant measurements based on this reflection technique have

consistently employed a mathematical approximation to Eq. (1). It is the

purpose of this present paper to examine quantitatively the errors inherent

in that approximation. For that purpose only , the assumption of a perfect

reflection at the dielectric-metal interface will continue to be employed in
this paper . To describe the approximation in question, in the limit 6 = TT, Eq. (1) reduces to a "minimum envelope"

-na\_ 2 s = J£z^ L (2)

(1 - pe

d

)

for n a half-integer, and a "maximum envelope"

-nttX, 2

L = JPJL^ > (3) -naX , 2

(1 + pe )

for n an integer. The index n = 8d/n designates successive minima and

maxima in the reflection coefficient profile and X = 2tt//? is the "wavelength"

d

within the dielectric of the incident radiation. The use of Eqs. (2) and (3)

to treat data presumably described by Eq. (1) we designate as the "perfect

square" approximation.

At the same time, this paper is intended to serve as an introduction

to subsequent papers of this series. The series as a whole is concerned with

improvements in the interferometric technique for measuring the complex

dielectric constant of liquids. Such improvements include a newly designed

apparatus as well as extensions of the theory and mathematical techniques.

In particular, consideration will be given to the true nature of the reflection

at the dielectric -metal interface.

2. THE APPROXIMATION 6 = TT The validity of the perfect square approximation depends upon the

degree to which 6 « Tt . For that reason, 6 values have been calculated

for a range of values of the complex dielectric constant using standard

expressions for the complex Fresnel reflection coefficient. For radiation

of free-space wavelength X = 2.143 mm and angle of incidence R = 35 , results of these calculations are shown in the contour diagram of Fig. 1,

where the phase shift 6 is given in degrees. This figure was hand-drawn

from computer -calculated 6 values and does not show fine details in the
shape of the contour lines.

From the 6 values indicated, it is clear that departures from

the relation 6 = V can be significant, particularly for high loss materials.

The extent to which Eqs . (2) and (3) then fail to describe the extrema of

Eq, (1) must then be determined. To illustrate this effect, we have

selected the arbitrary values e' = 2.50 and c" = 0.40, for which 6 » 175 , and calculated R vs d using Eq. (1), together with the corresponding

"envelopes" defined by Eqs. (2) and (3). Resultant curves, which were

both computer calculated and computer drawn, are shown in Fig. 2.

Even for the value V - 6 « 5 , these "envelope" curves fail to

fit the extrema of Eq. (1) by amounts greater than the expected experimental

error. It also seems worth noting that those points from Eq„ (1) which

most closely approach values defined by Eqs. (2) and (3) may occur at

various points relative to the peak of each extremum. The question of

precisely what power level should be fitted to Eqs. (2) and (3) in each

case then becomes quite ambiguous.

For a different set of values of c

1 and C M

, agreement between

Eq. (1) and Eqs. (2) and (3) may be much better. For example, curves

analogous to those of Fig. 2 using the values c'

r = 2.62 and €" = 0.28, for

which 6 ?« 178 , are shown in Fig. 3. In this case, the envelope curves

are tangent to the extrema of Eq. (1) to a much better approximation.

However, a comparison of Figs. 2 and 3 shows that the range of

dielectric thickness over which data points are selected introduces an

additional variable in the use of the perfect square approximation. In

Fig. 2, the envelope curves come closest to the extrema of Eq. (1) at

small d values and then deviate as d increases, while in Fig. 3 the

opposite behavior is found. Without prior knowledge of the complex

dielectric constant being measured, one cannot then specify the range

in dielectric thickness from which data points would yield the most

accurate determination.

If we also consider that range of c

1 and e" values in Fig. 1 for

which measurements using the perfect square approximation might be

carried out, the corresponding range in 6 values suggests that the
resultant errors may be less than or much greater than those of the

examples described. An adequate description of such errors then requires

explicit calculation, both with respect to a range of c

1 and €" and a range in

d over which extremum valued are employed.

3. THE DETERMINATION OF e

1 AND €"

Determinations of €' and e" using the perfect square approximation

require values for the dielectric thickness increments Ad between successive maxima or minima (Ad ^\j2), together with the maximum and

minimum power levels. In order to separate effects of the perfect square

approximation from experimental errors, we have calculated such quantities

directly from Eq. (l) for a series of values of €' and e". For this purpose, an iteration technique known as the Method of False Position was employed

(Appendix).

In practice, several of the minima of a given experimental curve are

not used in subsequent calculations because their amplitudes are too small

to be measured accurately. Which minima are deleted depends upon the

complex dielectric constant of the material being investigated. This practice introduces an additional experimental variable separate from the perfect

square approximation.

In analyzing this approximation mathematically, it would not be

feasible to guess which minima would be deleted in practice. In order to

treat this approximation more as it has been applied, however, we have

arbitrarily deleted the lowest minimum from each calculation. It will be

shown later that deleting such minima, whether in theory or practice, constitutes an additional source of systematic error when 6 7^ IT. The remaining amplitudes S

2 and L

2 and corresponding dielectric

thicknesses d .obtained by iteration, were then treated using the same tech- n

2

nique previously applied to experimental data. However, initial values for

the quantity a\ (used in calculating p ) were not estimated but were calcu2

lated explicitly from the e' and e" values used in each case. Values for p

were obtained from such aX values, rather than directly, in order to duplicate

d

the technique as previously employed. Resultant p values, together with the

calculated S

2 and L

2 values, were then employed in Eqs. (2) and (3), 2 rewritten in the form
x4 „. SA.

and

exp (-naX) = (p - S'V(1 - P s ) (4)

exp(-naX

d

)= (L

1/2 -p)/(l -pL

l/2 ), (5)

*/- 1 care being taken to insure that the S values had the correct sign.

Resultant quantities - ln[exp (-nttMjvs n were then fitted by a straight line

of slope a A., using least squares.

The slopes aX, so derived differ from the exact OiX values previously r d d

calculated by virtue of the perfect square approximation. Similarly, appliV. n values and calculated A. d d

A \

2 r /«\a 2 \

'a

cation of these derived a\ n values and calculated Rvalues in the equations

w£)2

['-(S

d

)

2

]

+ -\

and

A \ 2 aX,

calc ft) 7T

leads to complex dielectric constant values different from the €' and e"

values used as input in each case. Differences between the input €' and €"

and output c' , and 6" , values were then expressed as percentage errors

calc calc

A € ' and Ac".

4. NONLINEARITIES IN -In [exp (-nttXj] vs n

In experimental applications of the perfect square approximation,

systematic departures from the straight line -In [exp (-nO! X)] = nttX vs n

5

d d

are found such that points corresponding to the maxima lay above the line

while those for minima lay below the line. While this behavior was attrib5 uted to experimental error, we have found similar departures in entirely

theoretical calculations. Calculated curves which illustrate this behavior

for the values €' = 2. 6 and 6" = 0. 2, 0. 4, and 0. 6 are shown in Fig. 4.
5

In the experimental case, it was found that departures from the

mean line were most pronounced for low loss materials, disappearing as the loss increases. In Fig. 4, it can be seen that the opposite behavior is

found in a theoretical calculation. In spite of this difference, however, we

suggest that the origin of this behavior may be the same in both cases.

In order for these systematic departures to be observed experimentally,

it is necessary that (1) measurement errors must be smaller than the departure in question and (2) both minima and maxima must be included in these

calculations in order that oscillation about a mean line can be established.

In practice, both of these conditions preclude observation of these departures

in measurements on higher loss materials.

5

At the same time, it is true both experimentally and theoretically

that deletion of minimum points from the calculations leads to systematic,

positive errors in OL X , since the slope of the mean line is thereby increased.

It should also be noted, however, that this practice also leads to systematic,

positive errors in X,. 6

It has been shown earlier that wavelengths determined from spacings

between maxima are consistently long, while those from minima are consistently short. Deletion of minimum points will then yield net positive

errors in X,. This type of error differs from that previously described in

that it decreases with increasing n. In Fig. 4, and particularly on the curve

for e" = 0. 6, it can be seen that departures from the straight line increase

with increasing n. The effects of either of these errors would be minimized

if equal numbers of maximum and minimum points were included in the

calculations.

Examination of Eqs. (6) and (7) indicates that positive errors in both

aX-,a-nd \ lead to negative errors in e', while the resultant errors in €"

cannot be predicted without knowledge of the relative magnitudes of the

errors in a X, and X,. All of the errors discussed in this section, however,

d a constitute second-order errors. Of primary interest are the errors arising

from use of the perfect square approximation in the best possible manner.
5. ERRORS IN THE PERFECT SQUARE APPROXIMATION

Percentage errors arising from the perfect square approximation

were calculated for a range of values of e' and €" In determining €' , calc

and 6" . , all of the extrema in the range n =

l/z to n = 4, 6, or 8 except

calc

the lowest minimum in each case were employed. For each of these ranges

in dielectric thickness, the corresponding percentage errors in €' and 6 M

are plotted as functions of 6", for various values of €' , in Fig. 5. The £' values shown were chosen such that A€ ' increasing through the value

1% and A€" increasing through 2% would be illustrated. The curves were

drawn through points calculated at increments of 0. 1 in e".

From the figure, errors in both €' and €" are generally seen to be

positive and to increase with decreasing e' and increasing € ", i. e. , with

increasing loss tangent €"/€'. For e" values less than about 0.6, both

errors are less than 2%. For such materials, both errors are decreased

by using a greater number of data points. For higher loss materials,

however, the error decreases when fewer points are employed. In any

case, the differences in error arising from using different numbers of data

points are nearly as large as the errors themselves. For fixed numbers of

data points, the errors A€' and Ae" over a wider range of €' and €" are

also shown in the contour diagrams of Fig. 6.

6. CONCLUSIONS

It has been the purpose of this paper to provide the background for a more complete mathematical treatment of the reflection and interference

process by which the complex dielectric constant of liquids has been measured, as well as to demonstrate the need for such analysis. That such

analysis is needed is concluded on the basis that the perfect square approximation provides an inadequate test of the equation by which the interference

phenomenon has been described. The experimental demonstration of certain

anomalies in the use of this approximation is in itself evidence that the

approximation errors exceed the experimental errors. Having access to
computers, there seems to be no reason why the mathematical treatment

of such measurements should stand as a limit to their accuracy.

For purposes of estimation, on the other hand, if one does not

have access to a computer, the perfect square approximation does provide dielectric constant values to a limited degree of accuracy. By the

procedures given in this paper, such estimates may be made more

accurate. The errors calculated in this paper may in fact be employed

as corrections to the dielectric constant values obtained using the perfect

square approximation. At the same time, such estimates may be

employed as initial values in an alternative iteration procedure that will

be described in a subsequent publication.
APPENDIX: DETERMINATION OF THE LOCAL EXTREMA

The thicknesses d , n =

l/2 , 1, . . . , at which the extrema of Eq. (1)

n

occur, are determined from the expression

R'(d ) = 0, n = y2 , I, ... (8)

n

Since Eq. (1) is a transcendental equation, the sequence of solutions, [d },

is evaluated numerically. The first approximation to a given d is found

by scanning the positive d axis by increments of length (b- a) where b > a. When R'(a)R'(b) < there is at least one solution in the interval [a, b].

By choosing [ a, b ] such that

(b - a) < I

, (9)

it is possible to insure that only one solution is bounded by [ a, b]. Once the

7 8

first approximation is found, the method of False Position (Regula Falsi) '

is used to determine the final solution to any desired accuracy. Each local

extremum is then given by the coordinates

(d

n

, R(d

n

)). (10)

10
ACKNOWLEDGEMENT

The author wishes to thank Lynn M. Thiel for the programming,

application of the iterating technique, and other valuable contributions to

this work.

11
REFERENCES

1. W. E Vaughan, K. Bergmann, and C. P. Smyth, J. Phys. Chem. 65 , 94 (1961).

2. W. E. Vaughan, W. S. Lovell, and C. P. Smyth, J. Chem. Phys. j$6,

535 (1962).

3. V. M. Rao, Can. J. Phys. 4]_, 1679 (1963).

4. M. Schunzel and M. Stockhausen, Z. Angew. Phys. _2_1_, 508 (1966).

5. W. E. Vaughan, S. B. W. Roeder, and T. Provder, J. Chem. Phys. _39,

701 (1963).

6. C. Pine, W. G. Zoellner, and J. H. Rohrbaugh, J. Opt. Soc. Am. 4_9,

1202 (1959).

7. A. S. Householder,Principles of Numerical Analysis (McGraw-Hill Book

Company, Inc., New York, 1953), pp. 121-122.

8. D. Booth, Numerical Methods (Butterworths Scientific Publications,

London, England, 1957), pp. 148-149.

12
r-O

LO 6

<u

a

u

i—1 ro

CD 32

a *-< CJ 1^-

u> o

•r-l m

O

_" +>

ir>

<ii

C

on

a> ^ .2

C II

*" £ a cj o -S -3 T CJ +J -ri co ^^^~ <D • r-l ^ u £ rd r2 a

i—i

w

M-l

U ,—1

C <L> -J

M M-^ ._, ^ c C (TJ

—

_ r-l <d

V l/5 « u S >a/ ^ ?

CO ^ a

a lo

ro ,rJ r-l 1—1 ,—• -G 7? U CN1 a g< ,

0) c

-M II CJ

_

x o CD -u CJ > ^

<« (U

u ^

rC • ri +J c

+-> i. &D § s £ c

CJ <D C75 ^ o CD —

r-l CD

<1) >

-3 ^ co > ^ +> -m h rr) ,£ c ^ >

Con calc o w

•r-l

C

+> l—l

r>0 i—i

oJ Li3

.1-4

<Z> o^ OO CO LO ro OJ V

13
q:

14

a1 so

<du 0)

T3 OO

(0

w ^ W o +-> in

TJ 'O) -

ID

H ° « > <u o

&, CD CD

CD CD £

mh CD

g

c\j

CD • r-l
O O ro

C\J CO

^

co oj q -

<\i o ^

cvi

ro

= o o

vu CD <<

00

CMO

II

00

C\]

II

CO>l

PS

u

CC

15
5

4

o

o 3

€

l = 2.6

1

1 1 2

5 37 4

9 5

II 2 2 2 2 2 2 n

Figure 4: The least squares fit of the sequence of points

{ (-ln[exp(-n<A )] = na\ , n) } from the perfect

square approximation. e

' = 2. 6; g" = 0. 2,0.4,

0. 6. 16
r

... 9 00 u <u

<u

CO>.

VI

u^

VI

a; g 2*

^ CD 3

CO Sh h A VI o ~ Hig M fl —

« 3 ^

CD - w|

a °

uM .

•d > s

' CD U

coto £

C

VI

(U-HlM 3

i—

I

i—

I

U

CD

GO
A€"(%)

Figure 6: Contour diagram of the percentage errors A €

' and

A e

" in the e

', e

" plane tor three ranges of dielectric thickness.

18
Interferometric Measurements of the Complex

Dielectric Constant of Liquids

II. Experimental Apparatus

William S. Lovell

ABSTRACT

Design details of a free-space, millimeter -wave

interferometer are described. The apparatus radiates at oblique

incidence to an air-dielectric-metal configuration placed between

dissimilar horns. Errors arising from the quasi-optical nature

of this radiation are described, and quantitative data are presented which demonstrate techniques for minimizing such errors.

The techniques employed should have general applicability in

millimeter -wave measurements. For the particular instrument

described, the errors are shown to be of such magnitude that

experimental reflection coefficient profiles should conform to

an adequate theoretical description. A means for more accurate

and absolute measurements of the complex dielectric constant

of liquids is then provided.

19
1. INTRODUCTION

In the first paper of this series (I), errors arising from the use of

a mathematical approximation in treating free -space, interferometric

measurements of the complex dielectric constant of liquids were described.

It was concluded that such errors preclude an adequate test of the theory

previously used to describe the reflection and interference process upon

which such measurements are based.

The present paper describes an apparatus for measuring the complex

dielectric constant of liquids at millimeter wavelengths by reflection and

interference of electromagnetic radiation from an air-dielectric-metal

configuration. The origins of energy losses in the system are also treated.

Certain effects of a "quasi-optical" nature are also taken into account. An

analysis of these effects is given as a basis for the particular equipment

design to be described. This analysis includes quantitative data illustrating

the manner in which errors arising from such effects are minimized. These

data also show that for the instrument described, these effects will introduce

negligible error.

This instrument then provides a means of testing the theory upon

which its operation should be based, if the mathematical approximation

mentioned earlier is not employed. Upon obtaining an adequate theoretical

description of the reflection and interference process, together with the

necessary mathematical tools, this instrument then provides a means for

more accurate and absolute measurements of the complex dielectric constant.

2. APPARATUS

The largest component of the millimeter -wave interferometer consists

of a goniometer device which establishes a vertical plane of incidence for

the millimeter -wave radiation. A semicircular, 3/4-inch aluminum plate

of 24-inch radius is mounted on a base having three adjustable legs for

leveling purposes. To this plate are attached separate transmitting and

20
receiving arms which are loosely geared together and rotate about a common axis perpendicular to the plane of incidence. A diagram of this

system is shown in Fig. 1.

With the introduction of a horizontal sample well, positioning of the

transmitting arm then establishes the angle of incidence. Minor adjustment

of the receiving arm then places the detector horn at the corresponding

angle of reflection. The additional motional adjustments shown in the diagram are incorporated into the construction of the transmitting and receiving

arms themselves.

On each arm, the millimeter -wave components are mounted on a

slide operated by a precision lead screw which permits longitudinal motion

through a range of 12 inches. The distance between each horn and the

sample well is then adjustable over a range of approximately 10-40 cm,

At the same time, mounting of the millimeter -wave components on

each slide is accomplished through use of circular precision bearings of

5-1/2 inch diameter at each end. The polarization angles relative to the

plane of incidence for both the transmitting and receiving systems can then

be rotated through 360 degrees. The bearings at the horn end of each system were preloaded such that runout with horn rotation was less than 0.0005

inch.

The sample cell, sketched in Fig. 2, was constructed primarily of

aluminum and then zinc-plated after assembly. All joints connecting

components within which the dielectric liquid would be placed were hardsoldered. Critical dimensions such as those of the piston, cylinder, and

sample well were established by honing and replating followed by precise

measurement at 68 F.

To describe the operation of this cell, the liquid dielectric is placed

in a cylindrical reservoir within the cell through a fill port. A drain is

provided at the bottom of the cell. Surrounding this reservoir is a water

jacket connected through hoses to a temperature-controlled, circulating

water bath. A thermometer leads into the water jacket through the top of

the cell.

21
Above the reservoir, a reversible synchronous motor operating at

3 rpm drives a precision lead screw to force a cylindrical piston into the

reservoir . Liquid is then forced through a small orifice into the sample

well. The cross -sectional areas of the piston, reservoir, sample well,

and fill pipe are such that the ratio of downward motion of the piston to

increasing height of the liquid is 3.00008. The actual rate of increase of

the liquid height is established as desired through setting the gear ratio between the motor and the lead screw, which has a pitch of 1 mm. The total

range of travel of the piston is 9 cm. Allowing for a maximum dielectric

thickness of 1 cm, the size of the reservoir is such that sample volumes in

the range of 250-650 ml can be accommodated.

Concentric with the piston lead screw is a micrometer reading

directly to 0.01 mm for determination of the piston height. Geared to the

motor drive in a 3:1 ratio is a reversible shaft encoder registering 1000

counts per revolution to an accuracy of one part in 10 . The output of the

encoder leads to a 5-digit display module whose least count then represents

0.001 mm (1 micron) of dielectric thickness.

The cross section of the sample well may be described as a 10 cm

square with semicircles of 10 cm diameter added to two opposite sides. The

total depth of the well is one inch. At the bottom of the well is placed a silver-plated reflector approximately l/4 inch thick having the same shape

as the well. Beneath this reflector, the well is channeled to provide even

flow of liquid onto the reflector from all sides. The reflector is easily

removable for cleaning purposes, or for the insertion of additional spacers

to adjust its height. To minimize vibration waves at the liquid surface, the

sample cell, which itself weighs approximately 100 pounds when filled with

water, is placed on a vibration- damping mount weighing 87 pounds.

The long dimension of the sample well (20 cm) lies parallel to a line

through the thermometer well, piston axis, and fill port. In operation, the

cell is placed so that the plane of incidence of the interferometer bisects

the sample well through its long dimension. The fill port, piston drive,

22
and thermometer are then located in front of the area in which reflection

takes place and are thus accessible during operation of the instrument.

Operation of the system as a whole may be described by reference

to the block diagram shown in Fig. 3. Operating voltages for the klystron

are provided by a regulated dc filament supply and a regulated reflector and

beam supply, both operating from a line regulator. Attached to the klystron

is a water jacket through which cooling water is circulated. A thermostat

attached to the water jacket opens at temperatures above ambient to remove

power from the klystron power supplies and prevent operation of the klystron

without cooling water.

The klystron output, which is square -wave modulated at 1 kHz for

detection purposes, leads to standard millimeter -wave components designed

for the 90-140 GHz band. The power output of the klystron is approximately

120 mw. Through a ferrite isolator, the millimeter -wave radiation is led to a directional coupler where a -10 dB reference sample is taken off to a

crystal detector. The main portion of the radiation passes through a dielectric

attenuator and E/H tuner to a standard rectangular horn. For purposes to be

described below, a dielectric lens is attached to this horn. The lensed horn

is surrounded by microwave absorbing material.

On the receiving arm, the radiation reflected from the sample well

enters another lensed horn of different design, also surrounded by microwave absorber, and passes through an E/H tuner and wavemeter to a crystal

detector. The 1 kHz detected signal, together with the reference signal from

the transmitting arm, is fed to a ratiometer. A voltage corresponding to the

ratio of these signals, from which klystron noise has effectively been

eliminated, is then fed to a digital voltmeter.

A frequency divider circuit operating from the line voltage is used

to time the digital voltmeter readings. Such readings, together with the

corresponding dielectric thicknesses as indicated by the display module,

are fed to a digital recorder and constitute the experimental data for a given measurement of the complex dielectric constant. As a check on the

23
system, a chart recorder driven by a synchronous motor is connected in

parallel with the digital voltmeter to provide a graphical record of the reflected power vs dielectric thickness.

3. OPERATION

Application of this apparatus to a particular measurement requires

determination of the angle of incidence and wavelength X„ of the incident

radiation. The former is determined geometrically, while the latter is

obtained using the wavemeter. With such quantities known, there remain to

be determined the reflected power levels and corresponding dielectric

thicknesses to be used in the appropriate theoretical expressions.

However, the experimental data from the digital recorder must first

be placed on the same scale as the theoretical expressions. It will be shown

that the quantities necessary to define such scales cannot be measured with

sufficient accuracy. The manner of actually determining these quantities

will be described in a subsequent paper.

Dielectric Thickness Determination

In the course of a measurement, the dielectric thickness is increased

b/ driving a piston into the sample reservoir which then forces liquid into

the sample well by displacement. Initially, this liquid fills in the channeling

and the space between the reflector and the walls of the well. At a certain

position of the piston, depending on the volume of liquid in the reservoir,

liquid then flows onto the reflector from all sides.

Because of surface tension, however, the liquid does not form a smooth layer above the reflector immediately, but only after a minimum

volume of liquid has entered the well. The thickness of the dielectric layer

first formed will depend upon the nature of the liquid and on the temperature.

24
The theory for the measurement process is based on the model of

plane, parallel air-dielectric and dielectric -metal interfaces., Consequently,

only those reflected power levels that are recorded after the smooth dielectric

layer is formed constitute valid data. Evaluation of all subsequent dielectric

thicknesses then requires determination of the actual thickness d corresponding to the first such data point.

The quantity d may be estimated by determining the thickness reading

on the display module at which the liquid first begins to flow onto the reflector.

Alternatively, the dielectric thickness might be calculated from measurement

of the volume of liquid in the reservoir. However, such procedures cannot

be carried out to the accuracy to which a dielectric thickness difference can

be measured. As will be described in a subsequent paper, the accurate

measurement of such a difference permits an accurate determination of d

to be made mathematically.

A source of error in determination of the dielectric thickness by

measurement of the sample volume is evaporation. Of course, such evaporation must take place during the course of a dielectric constant measurement

as well. Evaporation may be reduced by surrounding the sample cell with an

air-tight enclosure and allowing the system to equilibrate before a measurement is begun. To describe such evaporation as does occur, it has been

2 assumed that the dielectric thickness decreases linearly with time t. On

that assumption, evaporation effects should be cancelled out if one averages

two curves of the reflected power R vs thickness d, one taken with increasing

d and the other with decreasing d.

Although such a procedure does double the number of experimental

measurements that must be made, the assumptions upon which it is based

seem reasonable. However, if one is to treat the equations describing the

reflected power R vs d explicitly instead of approximately, it becomes not

only possible but necessary to incorporate that assumption of a linear

evaporation rate into such equations.

25

kd
Since the dielectric thickness as determined by the synchronous

motor drive varies linearly with time, the evaporation rate may be described

in reference to d rather than t. An evaporation constant k in millimeters of

liquid evaporated per millimeter of change in dielectric thickness as indicated

by the display module may then be defined. This constant is measured using

assumed dielectric thickness values d taken from the display module.

a Taking account of the initial thickness translation d , the dielectric thickness

d may then be expressed as

d= (1 - K)d + d . (1) a

where the values d are those given by the digital recorder. In practice, a

since measurements are taken at equal intervals in d , only the first and

a

last of N values of d are recorded, and intermediate values are then cal- a

culated. Using preset switches incorporated into the display module, a final

d value may be preselected and used to shut off power to the synchronous

motor driving the liquid at the end of a measurement.

Power Level Determination

The assumption that a perfect reflection occurs at the dielectric -metal

3 2

interface suggests the use of a reflection coefficient defined as R= |E /E I , where E and E represent the amplitudes of the incident and reflected electric

vectors, respectively, and 0^ R < 1. The corresponding theoretical expres1-4

sion for R, given as Eq. (1) in I, is also normalized since it reduces to

unity for d = 0. The reduction of experimental data to this scale is accom3, 4 plished by the correspondence R <—»1, where R is the reflected power

measured at zero dielectric thickness.

However, energy may be lost by another means even if the reflection

is perfect. Some energy may not reach the receiving horn because of divergence

in the transmitted beam. Ohmic losses in the waveguide, reflections, and

scattering in the lenses may also reduce the power detected at the receiving

26
horn. All such actual energy losses may be described in terms of a transmission coefficient T, where < T < 1, and the reflection coefficient may be

redefined as R= T |E /E | . For a perfect reflection at zero dielectric

thickness, the power R. would then be described as R =TR , where R M M

is defined as the maximum power output of the transmitting system beyond

the attenuator „ Using the correspondence R «—»1, R becomes normalized

and describes the total energy transfer to the receiving system, including

both transmission and reflection processes.

Application of this scale in such a manner would require determination

of R (or R and T)„ Using the apparatus described by the block diagram in

Fig. 3, power measurements consist of voltages recorded from the digital

voltmeter „ Even in a direct measurement of the received power, the actual

magnitude of such voltages would depend, among other things, upon the

attenuator setting and the efficiency of the crystal detector on the receiving

arm. For the purpose of eliminating klystron noise from the power measurements, the apparatus includes an additional circuit consisting of the directional

coupler and a second crystal detector. The ratio of the power levels detected

in the two branches is then taken by the ratiometer, so that the attenuation and

crystal efficiency in the reference branch also affect the voltage levels actually

recorded.

The operation of the circuit as a whole then depends upon the relative

millimeter -wave power in the two branches, as determined by the attenuation

ratio between the attenuator and directional coupler; the relative detected

power as determined by the ratio of the crystal detector efficiencies; and the

final voltage reading as determined by the amplification factor of the ratiometer. In contrast to the actual energy losses described by T, these effects

represent apparent power changes, and their net effect could be adjusted to

any desired value for a given measurement of the complex dielectric constant.

For the purpose of relating the experimental record in such a measurement, consisting of the set of N voltages (R. | i = 1 , . . . , N} , to a theoretical

reflection coefficient, the effects described may be represented by a gain

27
factor G. The perfectly-reflected power at zero dielectric thickness would

then be represented c by R = GTR. ., where R„ now represents a voltage as M ° indicated on the digital recorder. Determination of R in this case would

require determination of R and the product GT. Assuming a perfect

reflection, R could be measured using the configuration shown in Fig. 1,

while GT could only be obtained by some disassembly of the millimeter -wave

components

.

Apart from the inconvenience of such a procedure, the value of the

product GT so obtained would have to remain constant throughout reassembly

and a subsequent dielectric constant measurement. Dependent as GT would

be upon the attenuator setting, tuning of the detectors and E/H tuners, temperature variations in the detector sensitivities, and ratiometer gain, the

assumption that GT would remain so constant could not reasonably be made.

For that reason, a means is required for determining GT using the configuration shown in Figs. 1 and 3.

By virtue of the factor GT, the voltage R and the values R. obtained

in a dielectric constant measurement would be described on an equivalent

basis provided that perfect reflection occurs at the dry plate. In the theory

for the reflection coefficient given in the following paper in this series,

however, that reflection is assumed not to be perfect. On the other hand, in

relating R and R. , the restriction to perfect reflections can be eliminated

since the theoretical expression for the reflection coefficient R(d) includes

the case R(0) for d= 0. Whether reflection at the dry plate is perfect or not,

the voltage R_ may be described as Rn = r„R(0), where we define r\ -GTRX ,. M

This formalism now places the experimental data R. and the

theoretical expressions R(d) entirely on the same scale. The transformation

required may be written as

R.= r R(d.), (2)

where d^ is the dielectric thickness corresponding to the voltage R.

,

and R

Q represents the special case for d = 0. By making use of that

special case, either GT or RM

could be determined if the other were known.

28
That step is not necessary, however, since the R. values may be used for

1

' the same purpose. Since the factors G, T, and R are not themselves of

intrinsic interest, all that is required is a value for the product V . As will

be described in a subsequent paper of this series, the R. values essentially

provide a set of simultaneous equations from which r„, as well as the d and

K. values described earlier, may be obtained incidentally in the course of

obtaining the value of the complex dielectric constant.

4. INSTRUMENTAL ERRORS

The description given so far of the apparatus and its operation,

together with the theory and mathematical procedures described in subsequent papers of this series, suffice to establish the manner in which complex

dielectric constant data are obtained using this interferometric technique.

However, the accuracy of such data will depend upon the degree to which

several sources of error originating in the apparatus have been minimized.

For the most part, these effects arise from the "quasi-optical" nature of the

millimeter -wave radiation and lead to extraneous variations in received

power as a measurement is made which cannot be described by a constant

such as Tn

. The remainder of this paper treats the origin of these effects in

detail and outlines the manner in which their resultant errors may be reduced.

The fundamental requirement for accurate dielectric constant data is

that the experimental record as taken from the digital recorder must incorporate no extraneous effects such that it cannot be described by the theory.

As will be shown in more detail in a later paper, demonstration that the

theory and experimental data are consistent is based upon a "goodness of

fit" criterion. The complex dielectric constant data obtained are then subject

only to random error, which can be treated statistically.

29
Standing Wave Effects

The presence of a standing wave pattern along the horn-cell-horn

propagation axis has previously been noted.2

' 3.This pattern arises from

reflection of millimeter -wave radiation at each of the horn-air interfaces.

In a dielectric constant measurement, introduction of the dielectric changes

the optical distance between the horns. As a result, a sinusoidal variation

in power of period Xn /2 becomes superimposed on the measured reflection

coefficient profile. Translation of the receiving horn by X /4 to shift the

standing wave pattern between successive measurements and averaging of

2, 3

the resultant curves has previsously been employed to eliminate this effect.

The amplitude of the standing wave pattern may be reduced by tuning

the klystron and adjustment of the E/H tuners on both arms. In the present

apparatus, this pattern is further reduced by surrounding the horns with

microwave absorber, thus reducing the reflection of radiation from one horn

back to the other. In addition, the lenses on each horn, attached for purposes

to be described below, may act somewhat as impedance matching devices to

reduce further the reflections at the horn-air interfaces. Through some

combination of these effects, reflection coefficient profiles measured on this

instrument at receiving horn positions separated by Xn /4 have been found to

be indistinguishable on an ordinary strip chart recorder. For this reason,

complex dielectric constant data are then calculated from a single experimental record.

In the absence of dielectric, however, a standing wave pattern can

still be detected, either by relative motion of the horns or by motion of the

reflecting plate in the cell. The purpose of the latter procedure and the

apparent absence of a standing wave pattern in the experimental record will

be discussed below.

30
Diffraction Effects

The influence of "diffraction effects" on this measurement technique

2-4

has previously been noted,, Diffraction occurs in any case in which the

millimeter -wave radiation traverses an aperture sufficiently small in terms

of wavelengths. Separate diffraction effects may be distinguished by refer- ence to the aperture involved. In the present apparatus, the transmitting

horn, receiving horn, and sample well comprise such apertures,

A certain diffraction pattern may then be said to exist in reference to the

horn-well-horn axis. In a dielectric constant measurement, however, that pattern can only be seen by virtue of a change in the geometry of the system in the course

of a measurement. Any constant difference in power arising from the diffraction

pattern would be incorporated in the constant T~ . Such changes occur only at

the sample well, through the introduction of dielectric; hence, it is in reference

to that aperture that di scernible diffraction effects must be described.

To describe the motions involved, it can be seen in Fig. 1 or in the

exaggerated sketch shown in Fig. 4, that the transmitting and receiving horn

axes cross at a single vertex. The propagation axis is colinear with

both of those axes, so that maximum power is received, only when the reflecting plane in the cell is at a height corresponding to that vertex. If the

reflecting plane is moved upward by a distance h, the point on the plane

originally coinciding with the vertex moves a distance h sin 8 transversely

and h cos 8 longitudinally with respect to each horn axis. The propagation

axis then moves a distance Zh sin transverse to the receiving horn axis

and decreases in length by 2h cos 8' . Both components of this motion are expected to affect the power level measured at the receiving horn.

For purposes of discussion, those components of motion may be

treated separately in their relation to the diffraction pattern. Because of the

relative size of the apertures involved, that pattern may be said to arise

primarily from horn diffraction, i.e., the millimeter -wave radiation appears

in a finite beam having a definite radiation pattern. Transverse motion of

that beam may be described as a beam displacement, and will be discussed

31
in the following section. Longitudinal motions, for which the dimensions of

the sample well may become relevant, may be treated more easily in terms

of a diffraction effect.

In a dielectric constant measurement, the plane defined by the

dielectric -metal interface is not moved and may be placed at the vertex of the

horn axes. The air -dielectric interface necessarily moves, however, and in

the manner shown in Fig. 4. That motion does not change the physical hornreflector -horn distance, but it does change that optical distance. For a

dielectric thickness d, longitudinal motion relative to the diffraction pattern

results from the replacement of air by dielectric liquid over a distance

2d cos 6 . It is this same motion, of course, which leads to the superposition onto the reflection coefficient profile of any standing wave pattern

present, as discussed in the previous section.

In order to examine the magnitude of any variations in power resulting

from the diffraction effect, it must be separated from those interference effects

which yield the desired reflection coefficient profile. This diffraction effect

could be simulated by placing an equivalent aperture centrally on a line

between facing transmitting and receiving horns and moving those horns

symmetrically along their common axis. Such a procedure is inconvenient,

however, and may be approximated by a series of measurements at fixed

horn-horn distances in which the aperture is moved along the horn-horn

axis. Using pyramidal horns of 1. 8 X 2. 2 cm aperture and an operating

frequency of 110 GHz, results of such a set of measurements are shown

in Fig. 5.To obtain these curves, a circular aperture whose diameter is

indicated under each set of curves in Fig. 5 was centered about the horn-horn

axis and then drawn from one horn to the other by a synchronous motor.

Simultaneously, the detected power at the receiving horn was led through an

amplifier to a strip chart recorder to yield the curves shown. Each aperture

was centrally cut in a one foot square aluminum plate, and it was assumed

that no millimeter -wave radiation escaped the outer boundary of the plate.

To minimize extraneous standing wave effects, all of each plate except for the

aperture itself was covered with microwave absorber on both sides. The

32
fixed horn-horn distance in centimeters for each of the four curves in each

set is indicated in Fig. 5. On logarithmic ordinate scales, these curves then

indicate the relative power detected at the receiving horn as a function of the

displacement Z of the aperture along the axis between the horns.

Although highly exaggerated, these curves should suggest the nature

of the diffraction effect occurring in the measurement process. The exaggeration arises from the fact that the distance corresponding to Z over which

dielectric liquid is actually introduced in a measurement is at most on the

order of 1 cm. Consequently, one is concerned only with a central 1 cm

segment of the Z-axis for that curve having the appropriate aperture size

and horn-horn distance. More precisely, one is concerned with the vertical

displacement, at the central point of these curves, between that curve and a similar curve for a horn -horn distance shorter by about 1 cm. For a 10 cm

iris, corresponding to the smallest cross -sectional dimension of the sample

well of this instrument, the curves suggest that the difference in question is

vanishingly small. At least in relation to the beam displacement effect discussed in the following section, this diffraction effect is then presumed to be insignificant. However, it may play a minimal part, as discussed in a

later section of the paper, in determining the optimum configuration for

operation of the instrument.

Beam Displacement Effects

Transverse motion of the reflected millimeter -wave radiation pattern

arising from vertical motion of the reflecting surface has been described as a beam displacement. In a dielectric constant measurement, that surface is the

air -dielectric interface. Only that portion of the incident beam actually reflected

at that interface is then displaced in the manner indicated. The relative

magnitude of that portion is determined by the Fresnel reflection coefficient

p e . Motion of that portion of the beam relative to the cross -section of the

receiving horn is expected to yield additional variation in received power.

As with the diffraction effect, measurement of this effect requires

separation from effects arising from the dielectric itself. This separation

33

2A\
may be accomplished by replacing the dielectric surface by a floating metal

plate and driving the liquid in the manner previously described.

Such "floating plate" measurements are affected by all three of the

processes previously described: diffraction, standing wave effects, and

beam displacement. However, it was shown in the previous section that

diffraction effects as defined herein are negligible. The standing wave

pattern may be treated as a periodic oscillation about a mean established by

the beam displacement. Such a mean line drawn through the oscillations in

experimental "floating plate" curves should then indicate the effect of beam

displacement alone.

To represent this effect mathematically, the variation in received

power as a function of the height h of the reflecting metal surface may be

2

described by a function r (h) . At an air -dielectric interface, the relative

magnitude of the reflected beam as actually received is then determined by

the product r(d)pe , replacing h with d. The corresponding power level is

2 2, , then given by p r (d)

.

To examine this effect, "floating plate" measurements were made at a frequency of 110 GHz using the pyramidal horns previously described.

2 Relative power levels r as a function of the height h in mm of the reflecting

surface for several angles of incidence are shown in Fig. 6, where the

indicated curves constitute mean lines drawn through the standing wave

patterns. These patterns were essentially constant in amplitude among the

curves taken. The ordinate scale for these curves was established using

the calibrated millimeter -wave attenuator. Since these curves were found

to be very closely symmetrical for ±h, i.e., for the reflecting surface above

and below the vertex of the horn axes (h = 0), only the +h portions of the curves

are shown.

These curves indicate that the beam displacement effect becomes more

4 pronounced for larger angles of incidence. This result is not surprising,

since it is shown in Fig. 4 that the transverse displacement of a ray should

be given by 2hsin0

Q

. To determine the extent to which this behavior is

34
exhibited by the actual millimeter -wave beam, an attempt was made to fit 2 2 2 the experimental curves using the function r (h) = -Kh sin , as shown

2 on the diagram. On the logarithmic ordinate scale, values for r (h) calculated

-2

at h = 2, 3, . „ „ , 7 mm for K = 3.55 and 3 . 85 mm are also plotted on Fig. 6 „ These results indicate that for the small range of transverse displacement

considered, the electric field intensity actually received varies with the

square of the displacement. The accuracy of the fit shown would not suffice

to justify use of that function as a quantitative correction factor. However, the

2 dependence of r (h) upon as indicated suggests that the interpretation of this

effect in terms of the transverse displacement shown in Fig. 4 is probably correct,

More importantly, however, the curves suggest the manner in which

beam displacement effects may be minimized. The smallest angle of incidence

consistent with other experimental requirements should be employed. Secondly,

one should employ a set of transmitting and receiving horns for which transverse displacement yields a minimum variation in received power.

As will be described in a later section, the lensed horns and configuration

employed -with this instrument were selected on the basis of these criteria. In a subsequent paper of this series, a more complete mathematical description of

beam displacement and related effects will be given. That analysis will suggest

the difficulties encountered in treating these effects mathematically, thus providing a rationale for the experimental procedures described in the present paper.

Multiple Reflection Effects

In a dielectric constant measurement, one additional effect tends to displace reflected energy transversely with respect to the receiving horn axis.

To describe this effect in terms of a ray model, an incident ray may be con- sidered to separate into a first-reflected ray and a transmitted ray at the

air -dielectric interface. The transmitted ray, or wave normal, is directed

at the real angle of refraction 9 towards the metal plate where it is reflected

at the same angle back to the interface. A second separation into reflected and

transmitted rays then takes place. The emergent ray lies parallel to the first

35
reflected ray but is displaced transversely by the amount 2d sin cos 6 , from the geometry. For subsequent multiply reflected rays, this displacement

process is repeated.

This guiding of energy away from the initial point of incidence serves

to conserve the tangential components of the energy momentum in the incident

beam, A more complete theoretical treatment of this process will be given

in a later paper in this series. For the present, it suffices to note that

multiple reflections lead to a transverse displacement of the reflected energy

similar in character to the beam displacement effect described earlier.

Unlike the beam displacement and other effects described earlier,

however, the multiple reflection effect cannot be separated experimentally

from the interference effects which yield the reflection coefficient profile.

Both types of effect require the presence of a dielectric layer and occur

simultaneously. The floating plate measurements illustrated in Fig. 6 do not

incorporate multiple reflection effects. No simple experimental estimate of

the magnitude of the multiple reflection effect can then be made. Evidence of

the magnitude of this effect can only be found in the degree to which the

experimental record in a dielectric constant measurement satisfies the

theoretical reflection coefficient profile.

To examine the magnitude of that effect qualitatively, the first displacement from double reflection will be less than the beam displacement,

i.e., Zd sin & , cos 6_ < 2d sin B' Secondly, the energy described by any

multiply-reflected ray is attenuated by passage through the dielectric, so that

the effect of its displacement will be reduced. In addition, for reflecting

surfaces above the vertex of the horn axes the displacements arising from

these two mechanisms occur in opposite directions, thus tending to cancel.

Finally, multiple -reflection effects are minimized by the same procedures

which minimize beam displacement effects, i.e., the use of small angles of

incidence and a large surface on the receiving horn.

36
With regard to multiple reflections, however, it also becomes necessary

to insure that those rays which eventually reach the end of the sample well

represent negligible energy. It is for this reason, among others, that the

sample well in this instrument is elongated in the direction of the plane of

incidence. For both displacement effects, the limiting factor with respect to

the reception of the total reflected energy is then the area of the receiving horn.

Error Minimization

Of the several sources of instrumental error described, the beam

displacement effect appears to be both the most prominent and the most

amenable to control. Standing wave and diffraction effects are minimized,

for the most part, in the basic construction of the instrument. Multiple

reflection effects, on the other hand, are minimized by the same procedures

which minimize beam displacement effects.

These procedures consist of insuring that as much as possible of

the energy reflected from the air-dielectric-metal configuration is actually

received. To accomplish that end for a given set of horns, a minimum angle

of incidence should be employed. The optimum set of horns will consist of a transmitting horn yielding a minimum beam width and divergence, and a receiving horn of large cross -section and a flat response. In basing the

instrument design on direct experimental evidence, however, the optimum

horn system was first selected, and then the behavior of that system with

respect to both the angle of incidence and the horn-horn distance was examined.

To select the optimum horn system, five combinations of three types

of horn were studied. These horns consisted of (A) a commercial pyramidal

horn designed for the 90-140 GHz band, (B) horn A with an attached plano-convex

lens, and (C) a commercial conical lensed horn of 3 -inch aperture designed

for the same band. The horns of type A were those employed in obtaining

the data illustrated in Figs. 5 and 6. The lens attached to obtain horn type B

was of 3 cm diameter, constructed from a commercial polymeric material

37
and designed from a standard lens formula. The lens was attached on its

flat side by a shallow notch fitting the end of the horn.

The five combinations of transmitting-receiving horns examined were

as follows: (a) A-A, (b) B-A, (c) B-B, (d) B-C, and (e) C-C. To examine

these horns, the transmitting and receiving systems were mounted in-line on an optical bench to provide direct transmission. For each combination,

measurements were then made of the variation in received power P with

transverse displacement £ of the receiving horn relative to the transmitting

horn axis. For four of these combinations, results of these measurements

at an operating frequency of 140 GHz are illustrated in Fig. 7.

The curves a, b, and c represent the effect of no lenses, a small lens on the transmitting horn, and small lenses on both horns, respectively.

Narrowing of the radiation patterns of both horns in this manner appears to

yield a simple narrowing of the response curve of the horn-horn system with

respect to transverse displacement. (The abscissa scale of Fig. 7 was

located only approximately, and appears to be offset from the transmitting

horn axis .

)

Curve d represents the effect of using a small lens on the transmitting

horn and a lensed horn of relatively large aperture on the receiver. From the

interpretation given earlier, this system should yield a more constant response

with respect to transverse displacement. Curve d indeed contains a relatively

flat segment. Additional effects appear at large displacements, however, as shown by the dip in curve d. The horn system C-C containing two conical

lensed horns yields a curve (not illustrated) with a very deep central dip,

rendering that system impracticable.

The horn combination B-C, whose response to transverse displacement

is shown in curve d of Fig. 7, was then chosen for use in this instrument. As

can be seen in that curve, there occurs an optimum range of transverse displacement over which the received power remains relatively constant. When

this horn combination is employed in the interferometer, there will then exist

an optimum height for the dielectric -metal interface. Starting from that

initial height, power variations arising from the beam displacement effect are

minimized.

36
To determine the optimum configuration of the instrument with respect

to the angle of incidence and horn-horn distance H, a series of floating plate

measurements was made at 140 GHz using the horn combination B-C. As

noted previously, such measurements are affected by diffraction effects,

standing waves , and beam displacement. As such, these measurements

indicate the degree to which such effects may constitute sources of error in

dielectric constant measurements.

For the values H = 30 cm and 40 cm and 8 = 30°, 40°, and 50°,

results of these floating plate measurements are shown in Fig 8, where the

ordinate scale is simply that of the recorder paper and has no absolute

significance. Except for the use of a different set of horns (and a higher

operating frequency), these curves were obtained by the same procedures used

to obtain the curves of Fig. 6.

In Fig. 8, however, the curves are reproduced just as taken, including

the standing wave patterns. Mean lines drawn through these patterns as was

done in obtaining Fig. 6 would in this case yield essentially flat, straight

lines. To that extent, beam displacement and diffraction effects have then

been eliminated as sources of error in this instrument.

The oscillations in Fig. 8 are identified as standing wave patterns on

the basis of their wavelengths. The lack of complete symmetry in some of

the curves suggests that other effects may also be occurring. The curves also

suggest that the amplitude of the standing wave patterns may be affected by the

angle of incidence and the horn-horn distance. A part of the differences in

amplitude may have resulted from changes in tuning of the millimeter -wave

components. In any case, the curves indicate that for the configuration

6 = 30° and H = 30 cm, the standing wave pattern is least in amplitude, and

symmetrical. That configuration was then adopted for use in this instrument.

39
5. SUMMARY

An apparatus designed for free-space, interferometric measurements

in the millimeter -wave region of the complex dielectric constant of liquids

has been described. Fundamental sources of error including vibration and

klystron noise have been minimized in the basic design of the instrument and

its associated electronics. A formal system for relating power and dielectric

thickness data in the experimental record to a theoretical reflection coefficient

has been given.

Additional sources of error arising from the quasi-optical nature of

the millimeter -wave radiation have been identified. In addition to standing

wave effects, there occur diffraction, beam displacement, and multiple

reflection effects. Experimental data which exhibit techniques for minimizing

these sources of error have been presented. The floating plate measurement

constitutes a means of testing the operation of the instrument and the magnitude

of such errors with an ease equal to that of a dielectric constant measurement.

Of the sources of error described, however, the influence of multiple

reflection and standing wave effects can only be determined from the degree

to which experimental data may be adequately described theoretically. To

the extent to which it is significant, the multiple reflection effect may tend to

cancel the beam displacement effect, if an appropriate initial height of the

dielectric-metal interface is chosen. The phase variations over successive

multiply-reflected rays may also tend to cancel standing wave effects.

That a measurable standing wave effect using a floating plate apparently becomes insignificant in a dielectric constant measurement, in which multiple

reflections occur, has been noted.

In subsequent publications, theoretical expressions describing the

reflection coefficient profile will be derived, the mathematical techniques

necessary to obtain the complex dielectric constant from such expressions

will be described, and a series of experimental data from which an error

analysis may be carried out will be presented.

40
ACKNOWLEDGMENTS

The author wishes to thank Lynn M„ Thiel for helpful advice on the

mathematical formalism employed, and Malcolm M. Anderson, Rudy Yoshida,

and Frederick E c Seiller for assistance on the experimental measurements.

41
REFERENCES

1. W. S. Lovell, Paper I.

2. W. E. Vaughan, W. S. Lovell, and C. P. Smyth, J. Chem. Phys. 36 , 535 (1962).

3. W. E. Vaughan, K. Bergmann, and C. P. Smyth, J. Phys. Chem. 65 , 94 (1961).

4. M. Schunzel and M. Stockhausen, Zeit. Angew. Phys. 21, 508 (1966).

5. M. Born and E. Wolf, Principles of Optics (Pergamon, 195 9), pp. 612-3.

6. A. F. Harvey, Microwave Engineering (Academic Press, 1963), p. 648,

42
O

ni .—

niO

•r-> 4-» Oa

-d

n3M

•i-i c!O

toO

Ona 

o

CD m

f-i CD

CD <D > >

ni ni i i u u

CD CO

+•> +-> (D CD a a s a

CD h

43
To

Display

Module

Synchronous

Motor

Thermom

Piston

Housing

Water

Output

Water

Jacket

Leveling

Legs

Shaft

Encoder

Gear Box

Micrometer

Sample

Well

Water Input

Base Sample Drain

Figure 2: Millimeter -wave interferometer: sample cell.

44
1 _i u_i zon

<

enz

<

a:

cr<

o

UlO

UJ

cr

1

O CC F* G o: —

-Q:

-0=

-I- <-> CO as

-<t

LbdJ

O

o

-*

L^J cr

C£ q- -Q:

o

• r-l

(L)

M

• r-l

rO O

CO

• • CO u ni

a) sd a

«» o -

4-> <D

£3 --1 H Oh

0) S

> nJ

rtj co

U 4-> CD fi

S fX

bfi

•r-i

45
\ 4 \ / <b° /

\ &.'* /^C- /

v^> V#<r / "* /

\ "^ /

/ ^

^ V

\ ° 7

•v. / i 1 -A A° 1 c\ / ^ o \ / o 
 \ / °

o \ /^

Figure 4: Geometrical model of measurement process showing

longitudinal and transverse motion of reflected ray.

46
7cm IRIS

40\

25

30

35

9 cm IRIS

d b

1

40

"!r ^^S ~rvi ^XJr^^o'<17

rr^

l — z

10 cm IRIS

Figure 5: Diffraction effects at 110 GHz for irises of 6-10cm diameter.

Curves show the variation in received power as the iris is

translated on the axis Z between the horns for the total hornhorn distance in cm indicated on each curve.

47
CM

ro

LO

CO o

CD

CMC

to CM

.c:

m

m

*: lO CD i ro ro M n ii *—

_c ^ ^

j= E + E o

CD oro

lOro

LO LO

LO oO

CO

bC

Fh

<u

•i-i o

XI CD

a id

-t-> ro fc T3 H ro <u X

05 bfl CO • i-i ro • id

•i-i id Vi >* o

(0 • i-l

bfi • r-i idro •H

ro3

<L> o

cri .i-i o

cr1 CD

J-i 3

id

u

CJ

CC]

CD

10 o

bO

XCO

CD X

H

d

-(->

-l-> CO Jd

C> (D nn

<1)

X 1 Tl

»—l ro o CD

+-I +»

CD

C

(h

CCi

U

n

1—12

ro

X sd o

i-H CO

CD i* ro

<ti ro o

CD

CDU

fn

u CO (ti 2

COro

<d

CD

-i->

C

•i-i i-H

CO s

CD> 3 CD X

CDa

CD £ J

4-> +j

H lo

u CD id 00 u

CD > CD

CO • CO

CO CDU

Oh

CD

id T3

5h id CD nj >• i-l +->ro T—

CDOd

ro

4->

U

CO

CD>

in

"IS ^ ro in

0* . (D °° > n

CD CO

.1-1 3O

ro

CM LO

CNJ LO

LO

LO

CNJ LO

LO

CNJI

CDU

bO

•i-l

48
1.0

0.9-

0.8

0.7

-2.0 -1.0

J? (cm)

1.0 2.0

Figure 7: Relative power P as a function of transverse

displacement i for four combinations of transmitting and receiving horns as described in the

text.

49
CSJ

ro

CO i co

tf

cu

i-i Tl Vh flh

pq • rj e

-- i o

•i-i 00 • cu i—

rt

r-" &

fl * Si rt

•i-i +j a

4-1 4-1 o o cu X! oocu

(1) >

COC

•t-i •i-i

1-4 u

(i1 S o

si

fi

•i-<

n

cu +->

a

Si

CO

+-> 00 o m

mm) s

using • r-l c

a.

CO

a

cu

!-4 0) m c

CD

(1) o

.,_,- +-> <U $3 _ ti Jn • r-J (!) C <u ?-l T3 T) + a

cu

U

CO

•i-i

.1-1 3 +-> -l-> 4-1 CO X rt n

nJ <D

CD -s-> i-l Q) a

0) r—

00

0) 4J

•+-> +->

PI cu TJ

.—t

t3 cu

(3

00 a)

f-l

co

a) rt o > O O CO ?n ri r—

cu 3 rtl

h T3 o 4->

00

CDh3

00 h

05

-C

CM

50
Interferometric Measurements of the Complex

Dielectric Constant of Liquids

III. Derivation of the Absolute Reflection Coefficient

William S. Lovell

and

Lynn M. Thiel

ABSTRACT

For measurements of the complex dielectric constant

of liquids by free -space reflection and interference of electromagnetic radiation, a theoretical reflection coefficient for the

air-dielectric-metal configuration is derived. The derivation

is based on a model which treats multiple reflections within

the dielectric explicitly. As an extension of earlier theory,

this reflection coefficient incorporates effects of (1) imperfect

reflections at the dielectric -metal interface, together with

other losses, and (2) radiation sources presenting nonplanar

wavefronts and having finite bandwidths . Effects arising from

a finite sample or receiving surface are also discussed.

51
1. INTRODUCTION

A previous paper (I) has shown that in applications of the free-space,

interferometric measurement technique, the complex dielectric constant of

liquids cannot be determined accurately using the "perfect square approximation" of the equation

R

,,. p - 2p cos (2Pd + 0)e +_e . "

1 9 ioRa M " 2ad x

2 " 4ad ' 1 - 2p cos(2pd - 0)e +p e

where the reflected power R is given as a function of the dielectric sheet thick- ness d. This perfect square approximation treats the phase shift 6 of electromagnetic radiation upon reflection at the air-dielectric interface as equal to

it. It was shown that the errors introduced by this procedure are unacceptable

except in estimating g*.

The ability of Eq. (1) to describe the reflection process adequately has not

been shown experimentally. An experimental test of Eq. (1) requires both an

apparatus sufficiently free of effects not described by a reflection coefficient

and an explicit treatment of Eq. (1) not involving mathematical approximation.

Such an apparatus, from which experimental data should conform to an adequate

2

theoretical treatment, has been described in the second paper (II) of this series.

In the present paper, certain modifications of Eq. (1) will be proposed. The

resulting expression incorporates Eq. (1) as a limiting case; hence, explicit

calculation using the proposed expression provides a test of the degree of

validity of Eq. (1). The actual manner of carrying out such calculations will be

described in the following paper in this series.

3-5

Recent derivations of Eq. (1) have been based on the use of boundary

conditions at the air-dielectric interface which incorporate implicitly the effects

of multiple reflections within the dielectric. These reflections are treated explicitly in the present paper in order to present more detailed description of the

reflection process. In addition, these recent derivations have employed, either

explicitly or implicitly, the assumptions that (1) there is no loss of power

in the experimental apparatus and (2) the incident radiation has a single

52
wavelength and is propagated at a single angle of incidence. In the present

paper, an alternative set of assumptions is employed: (1) the loss of power

in the system is appreciable and includes the effect of imperfect reflections

2

at the dielectric-metal interface, and (2) the incident radiation has a continuous distribution of wavelengths and is propagated at a continuous distribution of angles of incidence. Together with the explicit multiple -reflection

treatment, these modifications havebeen found necessary in order to describe

conditions existing in the experimental apparatus.

2. DERIVATION FOR THE IDEAL CASE

The air, dielectric, and metal media in Fig. 1 are designated by the

numbers 1, 2, and 3, respectively. At the coplanar interfaces between these

media, each reflection coefficient r.. and transmission coefficient t. . is denoted by selecting the proper subscripts corresponding to the media defining

the interface. The complex angle of refraction is represented by and the

i complex propagation vector by k. . Notice that -k = k = k = k = . . . , kd258 =k = k = k = . .. , and -k = k = k,= k = ... . Normal propagation within

d 3 6 V

the dielectric is then described by the complex propagation constant k cos . Amplitudes of the successive electric vectors in the air and dielectric

media are described by repeated application of the reflection and transmission

5 coefficients and represented bv the equations

E = r E

1 12

E * i = t i9 En^ r 9^ r 9i exp(-2kcos ed)]1"" 1 3n- 1 12 23 21 d d / 2

» n - 1 E 2 = t r E exp(-2k cos 6,d) [r r exp(-2k, cos d)] 3n 12 23 r d d 23 21 d d

E * xi = t n r t. E exp(-2k cos 9 ,d) [r r exp (-2k cos d)] n

, 3n + 1 12 23 21 d d 21 23 d d n = 1,2,... ,

where d is the dielectric thickness. The complex reflection coefficient is

defined as E_ oo E .

53
Since lr r exp (-2k cos 6 n d) I < 1, the infinite series in Eq. (3) converges

1 23 21 d d

absolutely and uniformly to a sum which after rearrangement becomes

^R r 12 + r 23

(t 12 ^1 " r 21

r 12 } 6XP (- 2k

d cos 9d

d> E

Q

= l-r

2l

r 23 exp(-2k

d co S e d

d)

(4)

Applying the principle of reversibility (which implies r = -r . and

tt-rr=l) and the definition

k n cos . = a + i# , (5)

d d

then Eq. (4) takes the form

E

R

r l2 + r 23 exp[-2d(a + i/?)]

E 1 + Tj r exp[-2d(a + i/S)]

' (6)

Let the reflection at the air -dielectric interface be described by the

c Fresnel reflection coefficient so that r = pe . If it is assumed that there

is no loss of power in the system, which implies in part that the reflection at

the dielectric-metal interface is perfect, then r _ = -1 and Eq. (1) is obtained

7 by defining R(d) = |E /E | „ If losses in the system are assumed, let

. , R (J r = ae , where ae

1 ^ is a complex reflection coefficient with the amplitude

a < 1, then in this case

2 o Z-,/9 j * i\ " 2ad . 2 -40?d

R

/_,\ .. p + 2a p cos (2o d + - lp)e + a e . .

l + 2ap cos (2pd - 0- jp)e +a p e

and R(d) is defined to be the theoretical reflection coefficient. It should be

noted that Eq. (7) reduces to Eq. (1) in the case that a = 1 and j/) = f . Parenthetically, it may be noted that if r = -r = -pe , i e„ , medium 3 in Fig.

Q

is replaced by air, then Eq. (6) reduces to the Airy formula.

54
3. THE FINITE SAMPLE AND/OR RECEIVING SURFACE

In evaluating the infinite series in Eq. (3), it is implied that the

reflected energy as measured will incorporate equally all of the electric

vectors E In principle, this situation could only be realized on condition 3n+l

that (1) the dielectric sample and reflector are infinite in length, (2) the

dimensions of the receiving surface parallel to the plane of constant phase are

infinite, and (3) the receiving surface has a uniform response over its crosssection. In practice, it is then necessary either to approximate these conditions

experimentally or to give an alternative treatment of the reflection coefficient.

In (II), the detailed mechanisms by which reflected energy may not be

received were discussed, and it was shown that a reasonable approximation to

these conditions may be achieved. The steps required to achieve that

approximation include (1) use of a sample well of sufficient area so that the

receiving horn constitutes the limiting factor with respect to reception of

reflected energy and (2) design of the transmitting horn, receiving horn, and

interferometer configuration such that the diffraction, standing wave, beam

displacement, and multiple reflection effects discussed in (II) are minimized.

For now, it is the intent to present a method by which failure

to receive all of the reflected energy may be described mathematically. As

such, this treatment is concerned only with the multiple reflection and beam

displacement effects.

With respect to multiple reflections, the transverse displacement be

2

tween the reflected rays k. , k , k , etc., shown in Fig. 1 is given by

2d sin 0' cos 6„. Such reflections within the dielectric yield a propagation of

d

energy tangentially to the dielectric surface. That such propagation must

occur by some means is apparent from the fact that the incident electromagnetic

energy propagates with momentum components tangential to the dielectric

surface. In typical derivations of the reflection coefficient using Maxwell

Q

boundary conditions , this energy transfer is not obvious since only the normal

component of the Poynting vector is treated.

55
With reference to Fig. 1, consider a receiving horn oriented at the

angle of reflection such that the ray k falls within one edge of its cross -

section, but some later ray, say k ,, just misses the opposite edge. Assuming

that the horn response is uniform over its cross -section, a reflection coefficient

which describes the resultant effect (in the ray approximation) may be derived

by truncating the series in Eq. (3) at the (n' - 1)— term. The ratio of that

10

sum to the sum of the infinite series will be identical to the ratio of the

corresponding reflection coefficient, which implies

R

, (d)/R(d)= |1 -[r

23

r 21 exp(-2k

d

cose

d

d)] n '| > [l - | r ^r^ exp( -2k

d

cos 9

d

d) | n '] 2.

(8)

It then follows that

l>R'(d)/R(d) > [1 -( (T pe~

2ad

) n '] Z

(9)

substituting for r and r as before.

For small cr and p and for large a , i. e., for weak reflections at the

dielectric -metal interface and high loss dielectrics, the effect of multiple

reflections will be minimized. The consequences of not receiving the rays

for which n > n' will be reduced since they will represent negligible energy.

Consistent with the discussion given in (II), multiple reflection effects are minimized at smaller angles of incidence, since n' is thereby increased.

Without prior knowledge of the complex dielectric constant, however, n' could not be caluclated, and in any case the calculation would be impracticable, as is shown below. For that reason, the experimental procedures

described in (II) are employed to take account of multiple reflections.

To describe a response distribution at the receiving horn, let r (d) = n

r(d -nx) be some distribution function for which x is defined in Fig. 2. The

quantity (d - x) represents the change in height of the reflecting surface

required to place a ray such as k at an equivalent position by simple reflection instead of refraction and reflection. For the ray k„ , , that same height

3n + 1

56
is given by (d -nx), where n is the ray index used previously, but now

^ n. From Fig. 2, it can be seen that

tan 9'

n x(d) = d (10)

tan Q

where is the real angle that the wave normal makes with the surface normal.

To account for the finite horn dimensions with greater generality,

consider a characteristic function Y , x defined by (n,,nj

: (n ,n ) (n) ~

) 1' 2 (0, otherwise

(11)

where n and n are the indices of the first and last rays received,

respectively. These indices are dependent not only on the dielectric

thickness but also on the horn-to-horn distance, dimensions of the horn,

Xn ,and 6.. Such dependence arises from the beam displacement as well

as the multiple reflection effects. The infinite series of Eq, (3) now

takes the form

n.

(12)

E

R

00 E

n=0

E

3n+1

E

X

< n

i> n2

)In

00 £ X

(n 1( nJ r n n=0 x c

2

n=n

E 3n+l

E

o

r n E n

2

E rn n=n

Success in finding a closed form for the series of Eq. (12) depends on

the form of the terms. Experimental examples of r have been given

in (II).

The experimental minimization of beam displacement and multiple

12

reflection effects requires construction of a receiving horn which is

large enough to approximate a horn of infinite dimensions with a flat

response. With reference to the experimental apparatus previously

2 described, this is the approach taken in this paper and is valid for the

range of dielectrics under consideration.

57
4. DERIVATION TO INCLUDE A DISTRIBUTION IN X AND

Instead of assuming that the incident radiation has the single wavelength X and is propagated at the single angle of incidence 6 , assume that

the normal point (6 n ,X ) gives the maximum of a continuous frequency function

defined over the set

s = {(e,X)|e - 1 £'e*e '+ £. 0* x < -}. (13)

Let x and y be continuously differentiable functions of and X, respectively,

and let (x,y) map S onto the extended real plane R*. Assume (x,y) is

normally distributed and, for the sake of generality, also dependently dis13, 14

tributed. The normal frequency function " of (x,y) is then given by

exp

W(x,y)=

2,

x - jUx

2(1 - t )[_\ ax /Hsirafa \ x/\ y/\ y

2tto a V1 - T 2

x y

(14)

where the parameters jU , a , U , a , and T are the mean and standard

x x ' y y

deviation of x, the mean and standard deviation of y, and the correlation

coefficient between x and y„ The variable (0, X) cannot be normally distributed

since S is not equal to R* as is the range of (x,y). There exist one-to-one

functions x and y on [ 6 - — , + — ] and [0,oo) respectively onto R* which

satisfy:

*<v!> = -• x(eo + f' = -•

and (15)

y(0) = - 00, y(00) = CO .

For simplicity, let

\X = x(9 ) =0 and U = y(X_) = (16) X • y

7

58
and for practicability let

x'(0 ) = 1 and y'(X

Q

) = 1. (17)

Also, since the radiating horn contains a plane of symmetry perpendicular to

the plane of incidence, let

x(e

Q

- A0) = -x(9 + A9) (18)

for all -— ^ AB < — where A0 = - 0„ Since the distribution of A may be

regarded as symmetric with respect to X._ for narrow bandwidths , also let

' y(X

Q

- AX) « -y(X

Q + AX) (19)

for small A X where A X = X - \,

Let w(8, X) be the density function for (9, \). Since the Jacobian

of x, y with regard to 8 and X is J - x'(B) y '(X) , then

w(9, X) = W[x(0), y(X)]x' (0)y'(X) . (20)

The theoretical reflection coefficient with the assumed distribution

17

(14) is obtained by applying the well known weighted mean formula. For

this purpose let f = f(9, X;d) be defined by the function on the right-hand side

of Eq. (7). Then since f is continuous and bounded and w is a frequency

function,

//f(9,X;d) w(9, X)d9dX tt

R(d) = — = f(9,X;d)w(e, X)d9dX (21)

// w(9, X)d9dX

"9

2

exists where JJw(9, X) d0dX = lj R(d) is defined to be the absolute reflection

s

coefficient.

59

^
Practical application of Eq. (21) requires explicit functions for x and

y. Possibly there exists a number of functions which satisfy conditions (15)

through (19) and have continuous first derivatives. However, for reasonable

functions, the distribution is expected to be insensitive to such functions for

proper choices of <x and cr . Two functions which might be considered are x y

tan (8 - 8 ) and \ ln(\X ).

5. SUMMARY

For interferometric , millimeter -wave measurements of the complex

dielectric constant of liquids in an air -dielectric -metal configuration, a theoretical reflection coefficient adequate to describe quantitatively the

experimental reflection coefficient profile is required. In deriving that

reflection coefficient, the occurrence of imperfect reflections at the dielectric-metal interface, as well as other energy loss mechanisms, must be

considered.

Explicit treatment of the multiple reflections occurring in the

reflection-interference process by which the dielectric constant is measured discloses a need for either (1) a specially designed system of transmitting and receiving horns to insure reception of all reflected energy or

(2) a mathematical description of the beam displacement and multiple

reflection processes by which energy would otherwise be lost. The first

of these alternatives was determined to be the more practicable. However,

a formalism is presented by which such effects may be incorporated into

the theoretical reflection coefficient.

A more realistic radiation source yielding a nonplanar wavefront and

having a finite bandwidth is considered in Eq. (21). For this, a normal distribution in (x,y) was assumed above. It was chosen because the distribution

of tan (6 - 8 ) and \ ln(XX ) have been observed to be approximately normal

and the normal frequency function has been extensively studied. It must be

noted, however, that with the appropriate parameters, use of almost any

distribution which is symmetric and monotonically increasing as (8 , \ ) is

approached is better than the assumption that the radiation has a single

68
wavelength X. and is propagated at the single angle of incidence O

. It might very well be that there exist other distributions which are closer to

reality than the one considered above. The solution of the reflection

coefficient expression given in Eq. (21) will be discussed in a later paper.

61
REFERENCES

1. W. S. Lovell, Paper I.

2. W S. Lovell, Paper II.

3. W. E„ Vaughan, K. Bergmann, and C. P. Smyth, J. Phys „ Chem. 65 , 94 (1961).

4. W . E„ Vaughan, Thesis, Princeton University (1960) o

5. Mo Yasumi, Bull. Chem. Soc. Japan 24 , 53 (1951).

6. W. Pfister and O. H. Roth, Hochfrequenztechn. U. Elektroak. 51, 156

(1938).

7. E„ Hille, Analytic Function Theory (Blaisdell Publishing Co., Inc.,

1965) Vol. I, pp. 102, 107.

8. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1959),

pp. 60-61, 322-325.

9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), p. 496

10. L. B. W. Jolley, Summation of Series (Dover, 1961), 2nd Rev. Ed., pp. 2-3,

ll.o Reference 9, p. 502.

12. That the loss of reflected energy can be significant, but can also be

eliminated by appropriate horn design, is apparent from a comparison

of Figs. 6 and 8 of Ref. 2.

13. G. U. Yule and M. G. Kendall, An Introduction to the Theory of Statistics

(Hafner Publishing Co., 1950), pp. 238-239.

14. P. G. Hoel, Introduction to Mathematical Statistics (John Wiley & Sons,

Inc., 1947), p. 150.

15. J. Edwards, A Treatise on the Integral Calculus (Chelsea Publishing Co.,

1954), Vol. 1, pp. 825-826.

16. W. H. Fleming, Functions of Several Variables (Addison-Wesley Publishing

Co., Inc., 1965), p. 175.

17. M. Fisz, Probability Theory and Mathematical Statistics (John Wiley & Sons,

1963), pp. 64-80.

62
C9

LU

O

O

X

LUO

CO<

or o 1 o• t-l U

-t-> o

<u .—

CD

.1-1 TJM• 1-1 rt

CIrt . d 1

•-I CD

cl

n o• 1-1 (-> • 1-1 o

•i-i u

rt

m

M-l

(I

CD o u

•ft

M-l <d u

cl rn GO d) rt <—i a

60 o rt M

+->

—

o rt

<u CD 1—

^i

CD

u >-h u

CO

Cl Cln O •f>

..-i +->

+-> rt o fn CD

i-H 3

GO

d) •l-l m 1h Cl

(1)

O

.—

u

a,

•t-< t—trt i—

+->

d CD s £

1—1

cdua

GO

63
2 Q?

\-

\- -z.

NSMI HOR <

cr

h- 3

iix

Xi

n5>

•w

x)oo

tJoa

i—iO

•pH

0)so

<U

64
Interferometric Measurements of the Complex

Dielectric Constant of Liquids

IV, A Numerical Method for Determining the

Best Complex Dielectric Constant

Lynn M. Thiel

and

William S. Lovell

ABSTRACT

A numerical method is presented for determining the best

complex dielectric constant €* = C

1 - ie" from an experimental record

obtained at millimeter -wavelengths from free-space, interferometric

measurements of liquids. The method of selected points is applied to

the reflection coefficient expression to obtain a set of simultaneous

equations which is solved for an initial vector solution using the

generalized Newton iteration function. The principle of least squares

is then applied to achieve the best fit between the reflection coefficient

expression and experimental record. The desired values of €' and €M

are included as components of a final vector solution of the best fitting

theoretical curve.

65

. -
1. INTRODUCTION

Previous methods for the determination of the complex dielectric

constant €* = C

1 - ic" of liquids utilizing free -space, interferometric

measurement techniques at millimeter wavelengths have not given

dependable results. In this series of papers, three steps have been

taken to obtain €* more accurately: (1) Methods which require approximations to an equation assumed to describe experimental results have

been abandoned. (2) The theory of the interferometric measurement

2 technique has been re-examined. (3) Effects occurring in the experi3 ment not described by the theory have been minimized.

A method of finding the best value for €* is presented in this

paper. The method consists of fitting a theoretical curve to experimental data. For radiation polarized perpendicular to the plane of

2 incidence, the function which gives the theoretical curve can be

written as

77 r f f °rr C

f(0. X;d) w(8, X) d0dX - R = , (1) ° Jo Je-9 -

o 2

where

2 „ ,-,„ , c ,v -2ad 2 -40td

flD n ,x P ± 2o- p cos (2fid + - ij)) e + q- e . . Uo> a; a; - -1 ——- - - U;

i . -, /^^, -, c ,\ -2ad, 2 2 -4ad

1 + 2 cr p cos (2(3d - - if)) e +crpe

and

w(e, \) = x y y • (3)

27Tcr cr \1 - T x y

The functions x and y are taken to be

x(6) = tan (6- 9 ) ; y(X) = X ln(XX

_1

) . (4) o o o

66
2 From the Fresnel reflection coefficient, p and 6 are defined by

(p -p) 2 + a

2

2<yp

P = 7^ T~I" ; 6

= tan ' 2 I 2 2 /

(5)

(P + (3) + a p - (CT+ (3 g g

where

a =^o_ {[(€'- sin

2

e)

2 +€" 2 ]^ -(€'-sin2 9)}' ; nT2

p =^o_ {[ (€'-sin2

9)

2 +€" 2 ]^ +(€'-sin2

e)f ; (6)

^2

P = p cos 9

g °

2

for p = 27T/X. The actual dielectric thickness d is defined by

d = (l-K)d + d . (7) a o

The left side of Eq. (1) is a function of the variables d and R, and

a contains as parameters the elements of the set & = { e', €", d ,9 , ° ° 2,3

X > cr, lb, K., cr ,o- , T, r } , all of which have been defined previously.

o x y o Some of the parameters of £2 cannot be estimated accurately by

direct observation and must be evaluated mathematically. For convenience, let there be M of these unknown parameters and consider

them to be the components of an M-dimensional vector v. The

solution of a set of equations containing v^is the vector solution CL , where the components of a are the values given to the M unknown

parameters to achieve the best fit between the theoretical curve and

the experimental data. The best fit is defined by using the principle

of least squares.

Although a set of equations containing y^ obtained by substitution

of data into Eq. (1) cannot be solved explicitly, the set can be solved

numerically. Iterative techniques which can determine a solution to

67
any desired number of significant figures by successive application of

an iteration function are employed. Derivations of two such iteration

functions followed by their applications and an estimate of the accuracy

of a will be presented.

2. THE ITERATION FUNCTIONS

M Consider the M-dimensional real Hilbert space E of the

vectors v*~ = (v. , . . . , v, ,) with the Euclidean norm

1 M

I v II = (v • v) - I

2 V

m=l

(8)

M The space E is also a complete metric space with the Euclidean

norm serving as the metric. Define the operator <p which maps a closed subset SCE into itself to be a contraction mapping. Then,

by the Banach Fixed Point Theorem, there exists a unique fixed

point Otf-S such that

(p(a) = a . (9)

The point OL can be found from the recursion relation

(k+l)_ , (k) v = (p(v ) (10)

«-— «~ —

(0) for v CS by taking the limit to get

OL- Urn v =limcp(v ). (11)

k-* oo k-+ oo

5 The mapping <p is classified as a one-point iteration function.

For practical applications, what is meant by convergence of

/ 00)

< v, > and what constitutes a solution must be stated clearly. Since

a is a limit point of <j v^ j

if and only if a. is a limit point of \v[

f for l^i^M, then by the Cauchy convergence principle, Eq. (11) holds

if and only if, given any €> 0, there exists an integer N. (e) such that

68
(m) (n) v. - v. < € (1 2)

i l whenever m, n ^ N. for l^i^M . Let N(c) = max iN.(c)l . Then

Eq. (11) holds if and only if, given e > 0. the inequality (12) is satisfied

whenever m, n ^ N for 1 ^ i ^ M. By similar reasoning, |v f is said to

converge in practice if for a specified € > >

(k+D (k) I ....

v. -v. < e (13)

1 1 I

for all l^i^M and for the number of iterations (k+1) ^K(e), where K

is some positive integer. The vector v is then taken to be the

desired solution 0/ .

Generalized Newton Iteration Function

Consider a system of nonlinear equations

F ( v) = 0, m = l M (14) m *" M where F .,..., F are real valued functions defined on E „ If the

M

function F mapping E into itself is defined by F = (F, , . . . , F ), ~* "— 1 m

then Eqg . (14) can be written as

F(v) = <K (15)

To find an explicit expression for tp of Eq. (10) by inverse interpolation , let each F have continuous first partial derivatives in m

some neighborhood of OL, N(o:), and assume that the Jacobian matrix

j y = r j

ij] = [^H (16)

J

is nonsingular in N(a) . If G is the functional inverse of F, then

a = G(0) . (H)

Expand the right side of Eq. (17) in a Taylor's series about uCE to

69
get

M dG (u) M M d G (u)

a = g (u) - Z —r~— u - + 7 2 Z r-ir- u- u - + ° • - < 18) m m ~- ., du. j 2 /-J .% du.au. i j j=l j i=l j = l i j

for l^m^M. Observe that

u = F (v) ; v. = G.(u) . (19) mm'*- ii"*-

From Eqs. (19),

M BF (v) M BG.(u)

du = T, —- m - dv. ; dv. = Y, —

1—— du. . (20) m . , ov. i l . , ou. l i=l i J=l J

J

Substitution for dv. into the first equation of (20) yields

M M 3F (v) 3G.(u)

^ = z z r~ -ir1- du - (2i) m /-J

1 J1 . ov. ou. l i=l J=l i j

J

which implies

M oF (v) oG.(u) M oG.(u)

V m ^

i ~ V T

i ** - A " Sv. ou. A mi ou. mi (22) i=l l j i=l J

where 6 is the Kronecker delta. If the elements of the inverse mj -1 -1

Jacobian matrix, J , are denoted by J. . , then by Eq. (22)

dG.(u)

= J. . (23) du. ij

since J J = I, the Mth order identity matrix. Substitution of Eqs.

(19) and (23) into (18) gives

M

a =v - Z J" . F.(v) + 0( II F(v) II ) (24) m m . , mi i -* -*« ~~

70
2 2 for l^m^M, where the remainder 0( II F(v) II ) is of order II F(v) II . 2 Truncating the right side of Eqs. (24) to eliminate 0( II F (v) II ) yields,,

using matrix notation, the iteration function

fpy = x~ j " ty £(^ * (25)

where <p , v, and F can be considered as column vectors. This is the

generalized Newton iteration function, GNIF, and has been shown to

7 be of second order.

Least Squares Iteration Function

To obtain a from an initial approximation v , consider a real

valued function F of the form

F(x, y;v) = (26)

which is to be fitted to the set of N observed points P = {(x , y ) n ° n n

I 1 ° o n = 1, . . . , N/ „ Each x or y is assumed to contain a random

n n experimental error which is independent of the errors in all other

coordinates of the points in P . Also, there exists a corresponding

o set of N adjusted or calculated points P= {(x ,y ) | n=l N} , To each x and y there is associated a weight w and w , n n x y

respectively, which is dependent on the confidence held in the accuracy

o ^ o

of that coordinate. Define the residuals X=x -x, Y-y - y > n n n n n n and V =v° - v for l^n^N and l^m^M„ Since all (x , y ) CP He

m m m n n

on the calculated curve described by Eq (26),

F(x , y ; a)= , n=l, .... N (27) n n *-» "~

which imposes N conditions on the elements in P. The principle of

least squares requires that

N

S = Y (w X

2 +w Y2

) (28)

, x n y n

n=l n

y n

be minimized with respect to the points in P restricted by the N

71
conditions imposed by Eqs . (27). The fit which yields the minimum

value of S, S . , is defined as the best fit. mm

For each (x , y )CP and v , Eq. (26) in general is not satisfied. n n o —

o o . o o ov Let F be some number such that F = F(x , y ; v ) for l^n^ N. n n n n "~ The conditions of (27) are made linear in the residuals by expanding

the left side of Eq. (26) in a Taylor's series about (x , y ; Ot) . n n "- Neglecting higher-order terms and using Eqs. (27) gives the reduced

conditions

M

F° = F

n X +Fn Y + Y. F

n V ,n=l N (29) n xn yn ^ . OL m m= l m

where the notation F means partial derivative of F with respect to t, evaluated at (x , y ; a) . - n n

"*- To find S . subject to the constraints of Eqs. (27) or equivalently

of Eqs. (29), the method of Lagrange multipliers is used. For this

let

M

g = [ F

n X +Fn Y + X F

n V ] - F°, n=l, ...,N (30) n xn yn '-',01. m n m= l m

and consider S to be a function also of the V 's with zero coefficients. m

Form the linear combination

N

'-', n n

n=l

where X. , . . . , X-j are arbitrary constants. A solution which minimizes

S will also satisfy the system of equations

Tx~ = °' TT— = °> ^7— =0, n=l,...,N (32) n n m m=l , . . . , M

and

gn

= 0. n=l,...,N. (33)

72
Observe that Eqs . (3Z) reduce to

2w X+XFn

= 0, 2w Y+XFn xnnx ynny = 0, n=l,...,N (34)

n n

and

N

Yj X F* =0, m = l,...,N, (35) n a. n=l m

respectively. If the residuals in Eqs „ (34) are solved to get

A F

n A F

n X = n x

; Y = 2_X_ , n = l N (36) n w n w

x y n n

where A = - — X > their values substituted into Eqs. (33), and the n Z n

coefficients of the A 's collected into the coefficients n

(Fn

) 2

(Fn

) 2

L = + X , n=l,...,N, (37

n w w

x y n n

then Eqs. (33) reduce to

M

F

n

= L A + V. F

n V , n=l,...,N (38) o n n *-*

, OL m m= l m

which can be solved to give

M

A = -Y—(F

n

- Yj F* V ), n=l,...,N. (39)

n L o , Ci m

n m=l m

Also, Eqs o (35) can be written as N

T!AFn =0, m=l,...,M. (40) '-'

, n Oi n=l m

Substitution of Eq. (39) into (40) yields after rearrangement

73
M

z

3=1

V F F

L a a - n=l —

L

N n n

V. = T, o a , m=l, ...,M. (41)

J n=l L

n n

In matrix notation, Eqs. (41) can be written as CV = B, where

the elements C . of the MXM symmetrical coefficient matrix, C,

are given by the quantities inside the brackets on the left side of

Eqs. (41), the elements B of the MX 1 matrix B are given by the

terms on the right side of Eqs. (41), and the elements of the MX 1 matrix V are V , . . . , V . If C is nonsingular, the residual vector

V can be solved for as V = C B, and the corrected approximate

solution is

v = v° - V = v° - C

_1

B , (42)

where v and v can be considered to be column matrices. The least

squares iteration function, LSIF, which depends on all of the points

in P , can now be defined by

o

tp (v) = v _ C

_1

(v)B (v) . (43)

The derivatives in Eq„ (29) were evaluated at the point (x , y ; 0?)

.

n n ""** For most practical applications of <p defined in Eq„ (43), it suffices

instead to evaluate these derivatives at the available point (x , y ; v )„ n n -•**'

3. DETERMINATION OF THE VECTOR SOLUTION

The solution of Eqs. (14) or (26) is dependent on the experimental

2 data. The data consists of a sequence of power level readings,

{ R I n = 1 , . . . , N} ; observed initial estimates of the parameters

n

r

in &; the total change in thickness, Ad, of the dielectric liquid; and

the index of some extremum, either a maximum or a minimum, of

an experimental record P^.. PAT constitutes the set P and is defined N N o by P = { (d , R ) n = 1, . „ . , N} , where the assumed thicknesses N a, n n

are given by

da,l =C; d

a,n

= d

a ,n-l*i£f' n = 2,...,N. (44)

74
Assuming that Eq. (1) describes the experimental record for the

proper values of the parameters in £2 , a can now be determined.

Analysis of the Integration

Evaluation of the function of Eq. (1) and its partial derivatives

requires integration of a real valued function g = g(9, X ; d ; v)

over an area A in the 9, X - plane, where g - T f • w of Eq. (1). For

obvious reasons the integration is performed numerically over an A

of finite dimensions. The basis and technique required for this

integration are now presented.

Numerical Integration for a Double Integral

Let f be a real valued integrable function defined on some

>pi

9

2 connected region of E . An approximate integral over a rectangular

area A is given by the formula

//'

n

l n

2

f(x,y)dxdy« 2 2 w..f(x.,y.) (45)

A- 1=1 3=1 1J X 3

where the w. . are the weights determined by "squaring" an appro1J 10

priate Lagrangian formula for integration with one variable . For

this, let L and L be n x 1 and n X 1 matrices, respectively,

whose elements are the weights w., i= 1,2 for the formulae

L

b. l

1 f.(x)dx« Yj w*f.(x.), i=l,2 . (46) a^ 1 i=i J

l J

The matrix of weights w. . are then given by the outer product of L ~ V 1 z and L , i.e.,by LL = w. I , where w. = w. w. . 2 1 2 ij ij i j To apply Eq. (45), let (b, - a.) = h. a- and (b n

- a) =h o

0" ^^ ' ^ 1 1 1 x 2 2 2 y, where h and h are sufficiently large so that w(9, X) « for

9(jlS = [a , b ] or X£J1S = [a , b ] . This is possible since w defined

75
in Eq. (3) is a normal frequency function of x and y. Then A is

defined by the Cartesian product S XS .

Differentiation Under the Integral Sign

The real valued function of Eq. (1) can be written as

F(d , R ; v) = ff g(6, X ; d ; v) d0dX - R . a,n n -* J J a,n «* n A

Application of the GNIF and the LSIF requires evaluation of the

partial derivatives F , , F_ , and F . Except for F„

d R v R , a, n n m n

differentiation of an integral expression is then necessary. Since

the derivatives are found analytically and the integration is performed numerically, it is desirable to have

3f a

(47)

3d Bd ffgdBdX =//-|f- <*6dX ; a,n a,n » a a,n

(48)

ff gd9dx=yy^- d9dx.

5F 5

dv dv m m A Am

If Eqs. (48) hold, then F and its partial derivatives can be obtained

by numerical integration of g and its partial derivatives. If A is a compact subset of E , B an open set of E , and g, along with its

partial derivatives, is continuous on the Cartesian product A X B =

1 2

{ (0, X; d ; v) | (0, X)CA, (d ; v)CB> , then it has been proven a,n /•*- a,n *" that Eqs. (48) hold and the partial derivatives are continuous.

Application of the Iteration Functions

Before applying the previous theory, it should be stated that all

of the conditions that have been imposed on the function (47), its

derivatives, parameters, and variables are satisfied. The assumptions that the Jacobian and coefficient matrices are nonsingular are

13 valid if neither the set of Eqs. (14) nor (26) is functionally dependent.

76
Functional dependence can be avoided by the proper choice of para^

meters to be included as components of v„ With all of these

stipulations met, 0> is determined in the following three steps.

Initial or Given Parameters

All of the parameters of £2 are to be given initial values . Since

functional dependence is to be avoided, only M of those parameters

are included as components of v The initial values given these M

parameters are only approximate and will be adjusted using the

GNIF and then the LSIF. The remaining parameters are treated as known constants and their initial values are final.

To determine initial values for €* and d , consider a set of K

o extrema, { (Rk. , d %) | i = 1 , . . . , K} and the corresponding array

of indices, {n^.. j i = 1 , . . . , K} , where n^. = -r for k = 1, 2, . . . . The

set of extrema can be determined to sufficient accuracy from the

experimental record using a scanning routine if Ad/(N - 1) is sufficiently

small. The corresponding values for the index array are then

obtained utilizing the index of the given extremum mentioned earlier.

1,2 An approximation ' to Eq. (1) which assumes ij) .= 6 = ff , X n = 2(d , -d . )/(nv. -nv.) for all i^j, and (0, X) not distributed

d a,kj a.k^ Kj *i

enables p to be solved for as

P =

'

f

- (-l)

2nki r? ,e'VXd

(49)

r

o

(-l)

2nki Rj,re'n

kiaXd

for l^i^K. Eliminating p between any two equations of the form

(49) for i ^ j yields the function f defined as

-(nk +nk )aXd

/ -nkiaXd -r*.u\d \ f(aXd

) = 1+SjO- e

3 + R (e + s^ J 1=0,(50)

where

77
R. = s_,

i i

2 p 2O

1 J K. -Y

S. - R

(51)

\ -1, extrema o£ the same type , ) -1, extrema both min.

for s , = i , ^ . and s = \ , 1 ( 1> otherwise 2 ( 1» otherwise

Equation (50) is solved numerically for OiX , which requires an initial

approximation z to z=o;X... z is found by scanning the positive

z-axis by increments of Az, starting at some small z > and letting

(0) = z 2p+l A + -T~ Az (52)

when f(z + p Az) f(z + (p+1) Az) ^ for some positive integer p.

The root QfX .. is determined accurately using the recursively-formed

d

iteration function

14

<p(z) =^ (z) , q

ire

<^

1 = z ; £($. ,(z))

1 J_1

£' (.)

J — 6( Jl t t t I (J •

(53)

(54)

The set of solutions obtained by considering various combinations of

15 extrema is then processed for outliers and the value for Ci\ is

d

taken to be the average of the remaining solutions. The estimate

for X is given by

V

2(1 -K)

t

tZ

i=l

d t, .-d . a, K - t + l a,i nK-t-fi-ni (55)

where t is the greatest integer of \K. The initial value for C* is

now obtained from

€' = ^ • 2. + sin Q (56)

78
The initial value for d is determined from the average of all o

&

D. =

1

n

ki Xd

a,k.

(57)

for l^i^K, excluding those that are outliers.

The remaining initial values are established more directly.

The initial values for , X , k, cr , and cr can be found experimentally,

o o x y

The initial value for cr lies between and 1 and is estimated from

previous experiments. The initial values for if) and T are close to

TT and 0, respectively. Finally, the initial value for T is taken to

be somewhat greater than the power level reflected from a dry plate.

The initial values for 8 , X , cr , K, and T so obtained are used in o o o determining the initial values for €* and d . o

Second Approximation to the Vector Solution

The method of selected points is used to obtain a better approximation to a from the initial approximation. A set of M equations is

defined by substituting M distinct points from P into Eq. (47) to get

F (v) = F(d , R m «" a,n n

J v), m= 1, . . . ,M (58)

m m

The inherent assumption here is that the points in P contain no

experimental error and that for the proper choice of v , the theoretical curve passes through the M selected points. The solution of

the set of equations (58) is found by applying the GNIF, using the

initial values discussed previously.

Let {d. f j = 1 , . . . , N, ,} be the set of 1NL , second approximations

-~j l J M M

to a found by considering N different sets of M points. The second

approximation CL , which is most representative of the entire record,

o e is then obtained in four steps: (1) Alia, for which their sequence of

iterates did not converge to a point in E are eliminated from the set,

(2) All a. which converge outside some symmetric neighborhood of

J

79
the initial approximation to Oi are deleted from the set. (3) All Oi which constitute outliers are eliminated. The vector a. is an

outlier if any of its components are outliers with respect to the

array of corresponding components of the set. (4) o^ is then taken

o

to be the average of the remaining Oi. in the set.

Final Vector Solution

The final solution Oi is the vector which yields the best fit between

the theoretical curve and the experimental record. This vector is

obtained from a by applying the LSIF. When the LSIF is used, every

point of P contributes to the determination of Oi. For the present application of the LSIF, equal confidence is

assumed in all coordinates of all points. Therefore, all weights can

be set equal to one and then Oi is the vector which minimizes the sum

of the squares of the residuals. This implies that all points of P

contribute equally to the determination of OL

4. ACCURACY OF THE VECTOR SOLUTION

The utility of a determination of c* from an experimental

record depends on the accuracy in the components of a . If & is

the true vector estimated by C£, then the accuracy may be described

in terms of the absolute error e - Oi - Oi . A measure of e is given by

i

1 m m mean

= [ (e ) 2

] 2

, m=l M (59)

where the mean denotes the mean value of the square of the difference

e . This is the average value that would be obtained if the vector

solutions determined from experimental records repeated a large

number of times under similar conditions were considered.

If the errors in the points of EL are uncorrelated and these

17

errors are small, it has been shown that approximately unbiased

2 2 estimates s of cr are given by m m ° '

80
s 2=

TT-T7 C

_1

» m = l,...,M (60) m N - M mm

where C. . is the i, ith element of the inverse of the coefficient matrix

-1 1J ~

C . The s are called the estimated standard errors of the com- m

ponents „ Once the s„ are determined, confidence intervals on the a can 111 m

1

be established. If the probability of an event E occurring is P(E)

where < P(E) <> 1 , p is a fixed percentage, and £ is the p percent

value of £ = (b - OL )/s for the N-M degrees of freedom, then m m m

appr oxim ate ly

P(a - £ s <a <Q; + £ s ) = i .-£_ (61) m p m m m p m 100

19

for Um^M, The values C have been tabulated for various p

P

and degrees of freedom. Equations (61) say that there is the

probability 100 - p% that the two random variables OL ± £ s will m J p m

take on values which together include the unknown but true value

Oi . Then the intervals (ol -Cs , OL +£s ),m = l,...,M m m p m m p m

are the 100 - p% confidence intervals for the OL , the limits of the m

intervals are confidence limits for the OL , and the corresponding m

confidence is 1 -p/100. Also there exists a 100 - p% chance that the

magnitude of the absolute error in the mth component of OL is smaller

than f s p m

5. DISCUSSION

Although it is not possible to predict when a sequence of iterates

obtained using one of the iteration functions previously discussed

will converge, some observations are possible. Convergence will

not occur if:

(1) One of the functions in Eqs. (14) or (26) or its partial

derivatives becomes unbounded. In most cases the evaluated functions of Eqs. (14) or (26) must become small

for convergence.
(Z) The functions of Eqs. (14) or (26) are functionally

dependent„ Dependence can be caused by the inclusion

in v of parameters in £1 which are interdependent „ For

example, application of the above technique has shown

that any subset of { 6 , \ , cr , lb , /c } consisting of o o

^ three or more elements is interdependent. When functional dependence exists, the determinant of the

Jacobian or coefficient matrix approaches zero and

the norm of each iterate increases without bound.

(3) The vector v contains a large number of components.

Convergence is improved as the number of components

of y^is decreased.

(4) A large neighborhood of a is necessary to include

the initial approximation to 0! . In general, the closer

the initial approximation is to a, the greater the

probability for convergence. The vector composed

of the inital values of the parameters described above

is usually close enough to cc to yield a number of con- vergences per record using the GNIF. The vector^

is usually sufficient for convergence using the LSIF.

(5) The theory does not describe experimental

conditions. This gives an important method of checking

theoretical assertions, since the accuracy of the solution

is dependent only upon the agreement between the theoretical description and the experimental record.

To evaluate €* in practice, the five parameters C

1 , €", d , cr , o and r are usually included as components in v, hence their initial

values are only approximate. However, there are other parameters

which may be known only approximately that are not included in v. Errors in these parameters are absorbed by the values of some of

the parameters in v. This compensation is possible because there

exists interdependence between some parameters included in v and
those not included. Since g

1 and c

" are observed not to be dependent

on any other parameters, e * is unaffected by this compensation.

The second approximation to a using the GNIF is determined for

two reasons. First, a number of a . are attempted for each experimental record and the chance of obtaining at least a few convergences

is good. Then since a^ is usually closer to a than the initial approximation, the probability of achieving convergence in subsequent use

of the LSIF is increased. Secondly, convergence with the GNIF is,

at least as empirically experienced in this application to the complex

dielectric constant of liquids, much faster than with the LSIF. The

reduction in the size of the neighborhood of jv necessary to contain a_^ substantially reduces the number of iterations performed in obtaining a using the LSIF. ~~

(k)

If the partial derivatives of g(0, X.;d ;v ) are close to the

(k+l) a ' n "" partial derivatives of g(6, X.;d ;v_ ) the cost of computation per

a , n

iteration using the LSIF can be decreased by not calculating new

derivatives at each iteration. If new derivatives are not evaluated,

then the coefficient matrix C remains the same and only the matrix

B must be re -evaluated. Computation costs may also be reduced by

not considering the distribution in (6, X.) until the last iterations with

the LSIF.

To demonstrate the relative agreement between the theoretical

curve and the experimental record, consider the measure A defined

1 N

by A(N; a) = — 7 I F(d , R ; a) \ , where F is defined in n = 1 Eq. (47). A is the average difference in power level between the

theoretical curve and the experimental record. The size of A

depends on (1) the cardinality of P , (2) the size of the random

error in the coordinates of each point in PKT , (3) the number of

components of v, and (4) the agreement between theory and experiment.

Application of the techniques discussed in this paper necessitates

use of a large digital computer. The routine used here with Eq. (1)

83
was designed for more general application. In fact, any function

satisfying the conditions imposed herein can be solved for a vector

solution of any dimension using an arbitrary number of experimental

points, providing also that the sequence of iterates converges.

84
APPENDIX: AN EFFICIENT MATRIX EVALUATION

Let P be an M X M nonsingular real matrix and Q be an M X 1 -1

real matrix. The evaluation of a matrix T of the form T = P Q is

required for the practical application of Eqs. (14) and (26). An

efficient computation of T is accomplished by forming the M x (2M+1)

augmented matrix [I, P, Q ], where I is the M X M identity matrix,

and then reducing the submatrix consisting of the last M columns to

I by performing elementary row operations on the entire augmented

matrix. The resultant matrix is [ T, P , I] , where T is the

desired matrix. This procedure also yields the diagonal elements

necessary for evaluating the estimated standard errors in Eq„ (60^

For details on the reduction of the augmented matrix see reference 21
REFERENCES

1. W.S. Lovell, Paper I.

2. W.S. Lovell, L M„ Thiel, Paper III.

3„ W.S. Lovell, Paper II.

4. I. P„ Natanson, Theory of Functions of a Real Variable (Frederick

Ungar Publishing Co , I960) Vol. II, pp 216-219.

5. J.F. Trabu, Iterative Methods for the Solution of Equations , Prentice-Hall, 1964) p. 8.

6. Reference 5, pp. 218-220.

7. Reference 5, p. 221.

8„ T.M. Apostol, Mathematical Analysis, (Addison-Wesley Publishing

Co., Inc., 1957) pp.152-156.

9. M„ Abramowitz and I. A. Stegun, Handbook of Mathematical Functions

(National Bureau of Standards, Applied Mathematics Series 55,

Fifth Printing, August 1966, with corrections) p. 892.

10. R.A. Buckingham, Numerical Methods, (Pitman Publishing Co.,

1957) pp.496-4?9~!

11. A table of explicit derivatives is available upon request.

12„ W.H. Fleming, Functions of Several Variables, (Addison-Wesley

Publishing Co. , Inc., 1965) pp. 197- 199.

13. P.F. Franklin, Methods of Advanced Calculus , (McGraw-Hill

Book Co., Inc., 1944) pp. 60-63.

14. Reference 5, pp. 165-166.

15. W. J„ Dixon, in A.E. Sarhan and B.G. Greenberg, Contributions

to Order Statistics (John Wiley and Sons, Inc., 1962) Chap. 10H,

pp. 299-342.

16. W.E. Deming, Statistical Adjustment of Data , (John Wiley and Sons

,

Inc., 1948) p. 52, 138.

17. J.R. Wolberg, Prediction Analysis , (D. Van Nostrand Co. , Inc. , 1967) pp. 54-62.

18. H. Cramer, The Elements of Probability Theory , (John Wiley &

Sons, Inc., 1955) pp. 198-211.

19. Reference 18, p. 274.

20. Listings of the Computer programs written in FORTRAN are

available upon request.

21. J.B. Scarborough, Numerical Mathematical Analysis , (The John

Hopkins Press, 1966) pp. 284-297.

86

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